Analiza porównawcza ryzyka w modelach klasy GARCH
Aleksandra Lubicka
5 lipca 2019
Wstęp
Celem badania jest analiza porównawcza ryzyka rozumianego jako oszacowanie funkcji warunkowej wariancji w modelach klasy GARCH. W tym celu wybrano dwa modele z rodziny GARCH- …. Zbudowano portfel składający się z czterech kryptowalut, w którym każdego dnia udziały procentowe każdej z nich są proporcjonalne do ich kapitalizacji rynkowej oraz przeprowadzono analizę porównawczą oszacowań funkcji warunkowej wariancji w okresie in-sample oraz oszacowań wartości narażonej na ryzyko uzyskanych za pomocą rozpatrywanych modeli GARCH w okresie out-of-sample.
Dane modelu
Dane potrzebne do przeprowadzenia badania pochodzą z serwisu https://coinmarketcap.com/. Wybrano do analizy notowania kursów: tron, cosmos, neo, waves, nano. Próba pochodzi z okresu od stycznia do sierpnia.
Import i wstępna obróbka danych
library(lmtest)
library(xts)
library(RCurl)
library(rlist)
library(urca)
library(fBasics)
library(dygraphs)
library(tseries)
library(car)
library(rugarch)
library(fGarch)
library(timeSeries)
library.path <- .libPaths("/Library/Frameworks/R.framework/Versions/3.6/Resources/library")scrapData <- function(x){
# function to scrap the OHLC data from
# www.coinmarketcap.com
# load packages
library(XML)
library(RCurl)
library(rlist)
# set locale to English
Sys.setlocale("LC_TIME", "C")
# set url
theurl <- getURL(
paste0("https://coinmarketcap.com/currencies/",
x,
"/historical-data/?start=20130428&end=21000101"),
.opts = list(ssl.verifypeer = FALSE))
# read html source
tables <- readHTMLTable(theurl, stringsAsFactors = FALSE)
# clean the list object
tables <- list.clean(tables, fun = is.null, recursive = FALSE)
# unlist the list
table <- tables[[1]]
# rename variables
table$Open <- table$`Open*`
table$`Open*` <- NULL
table$Close <- table$`Close**`
table$`Close*` <- NULL
# convert characters to numericals
table$Open <- as.numeric(gsub(",", "", table$Open))
table$High <- as.numeric(gsub(",", "", table$High))
table$Low <- as.numeric(gsub(",", "", table$Low))
table$Close <- as.numeric(gsub(",", "", table$Close))
table$Volume <- as.numeric(gsub(",", "", table$Volume))
table$MarketCap <- as.numeric(gsub(",", "", table[, "Market Cap"]))
table[, "Market Cap"] <- NULL
# convert the date from chr into Date
table$Date <- as.Date(table$Date, format = "%b %d, %Y")
# sort the data
table <- table[order(as.numeric(table$Date)), ]
table <- table[, c("Date", "Open", "High", "Low", "Close",
"Volume", "MarketCap")]
return(table)
}
####################################################################################################
# function call
library(tidyverse)## Registered S3 methods overwritten by 'ggplot2':
## method from
## [.quosures rlang
## c.quosures rlang
## print.quosures rlang## ── Attaching packages ─────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──## ✔ ggplot2 3.1.1 ✔ purrr 0.3.2
## ✔ tibble 2.1.3 ✔ dplyr 0.8.1
## ✔ tidyr 0.8.3 ✔ stringr 1.4.0
## ✔ readr 1.3.1 ✔ forcats 0.4.0## ── Conflicts ────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ tidyr::complete() masks RCurl::complete()
## ✖ dplyr::filter() masks timeSeries::filter(), stats::filter()
## ✖ dplyr::first() masks xts::first()
## ✖ dplyr::lag() masks timeSeries::lag(), stats::lag()
## ✖ dplyr::last() masks xts::last()
## ✖ dplyr::recode() masks car::recode()
## ✖ purrr::reduce() masks rugarch::reduce()
## ✖ purrr::some() masks car::some()# tron, neo, waves, nano:
tron <- scrapData("tron")## Warning in scrapData("tron"): pojawiły się wartości NA na skutek
## przekształceniasummary(tron)## Date Open High
## Min. :2017-09-13 Min. :0.001419 Min. :0.001831
## 1st Qu.:2018-03-12 1st Qu.:0.020352 1st Qu.:0.021185
## Median :2018-09-08 Median :0.025236 Median :0.026068
## Mean :2018-09-08 Mean :0.030960 Mean :0.032981
## 3rd Qu.:2019-03-07 3rd Qu.:0.037311 3rd Qu.:0.039448
## Max. :2019-09-04 Max. :0.224499 Max. :0.300363
##
## Low Close Volume
## Min. :0.001091 Min. :0.001427 Min. :2.648e+04
## 1st Qu.:0.019608 1st Qu.:0.020355 1st Qu.:1.064e+08
## Median :0.024272 Median :0.025242 Median :2.107e+08
## Mean :0.028860 Mean :0.030956 Mean :3.321e+08
## 3rd Qu.:0.034994 3rd Qu.:0.037208 3rd Qu.:4.603e+08
## Max. :0.176712 Max. :0.220555 Max. :4.089e+09
##
## MarketCap
## Min. :1.860e+07
## 1st Qu.:1.392e+09
## Median :1.689e+09
## Mean :2.083e+09
## 3rd Qu.:2.488e+09
## Max. :1.450e+10
## NA's :15tron %>% as.tibble %>% tail## Warning: `as.tibble()` is deprecated, use `as_tibble()` (but mind the new semantics).
## This warning is displayed once per session.## # A tibble: 6 x 7
## Date Open High Low Close Volume MarketCap
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2019-08-30 0.0155 0.0158 0.0153 0.0156 439318790 1040672169
## 2 2019-08-31 0.0156 0.0157 0.0153 0.0156 467796844 1037558304
## 3 2019-09-01 0.0156 0.0157 0.0154 0.0156 424372234 1041912671
## 4 2019-09-02 0.0156 0.0161 0.0153 0.0159 458500298 1061853982
## 5 2019-09-03 0.0159 0.0161 0.0157 0.0159 422623093 1063540664
## 6 2019-09-04 0.0160 0.0160 0.0155 0.0156 407752910 1039747569neo <- scrapData("neo")## Warning in scrapData("neo"): pojawiły się wartości NA na skutek
## przekształceniasummary(neo)## Date Open High
## Min. :2016-09-09 Min. : 0.0802 Min. : 0.08521
## 1st Qu.:2017-06-08 1st Qu.: 1.4300 1st Qu.: 1.71000
## Median :2018-03-08 Median : 11.6800 Median : 12.12000
## Mean :2018-03-08 Mean : 24.5686 Mean : 25.93194
## 3rd Qu.:2018-12-05 3rd Qu.: 32.4900 3rd Qu.: 34.13500
## Max. :2019-09-04 Max. :187.9700 Max. :196.85000
##
## Low Close Volume
## Min. : 0.07229 Min. : 0.08018 Min. :1.560e+02
## 1st Qu.: 1.30000 1st Qu.: 1.53000 1st Qu.:8.534e+06
## Median : 11.16000 Median : 11.69000 Median :9.640e+07
## Mean : 22.95233 Mean : 24.56408 Mean :1.484e+08
## 3rd Qu.: 30.29000 3rd Qu.: 32.28000 3rd Qu.:2.158e+08
## Max. :151.46000 Max. :187.40000 Max. :1.664e+09
##
## MarketCap
## Min. :5.292e+06
## 1st Qu.:3.672e+08
## Median :8.342e+08
## Mean :1.630e+09
## 3rd Qu.:1.999e+09
## Max. :1.218e+10
## NA's :46neo %>% as.tibble %>% tail## # A tibble: 6 x 7
## Date Open High Low Close Volume MarketCap
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2019-08-30 8.77 8.92 8.69 8.76 214238128 618155575
## 2 2019-08-31 8.76 8.9 8.63 8.82 216947467 622362487
## 3 2019-09-01 8.82 8.89 8.69 8.8 202957135 620552317
## 4 2019-09-02 8.8 9.23 8.77 9.15 227666511 645598698
## 5 2019-09-03 9.15 9.37 9.09 9.3 241655994 655888043
## 6 2019-09-04 9.3 9.31 9.02 9.09 230107282 640861597waves <- scrapData("waves")## Warning in scrapData("waves"): pojawiły się wartości NA na skutek
## przekształceniasummary(waves)## Date Open High
## Min. :2016-06-02 Min. : 0.1275 Min. : 0.1515
## 1st Qu.:2017-03-26 1st Qu.: 0.4298 1st Qu.: 0.4512
## Median :2018-01-17 Median : 2.3800 Median : 2.4950
## Mean :2018-01-17 Mean : 2.8349 Mean : 2.9856
## 3rd Qu.:2018-11-10 3rd Qu.: 3.9275 3rd Qu.: 4.1625
## Max. :2019-09-04 Max. :16.1400 Max. :18.0700
##
## Low Close Volume
## Min. : 0.1227 Min. : 0.1272 Min. : 4912
## 1st Qu.: 0.4140 1st Qu.: 0.4298 1st Qu.: 303484
## Median : 2.2900 Median : 2.3800 Median : 7525894
## Mean : 2.6693 Mean : 2.8322 Mean : 13599345
## 3rd Qu.: 3.6600 3rd Qu.: 3.9175 3rd Qu.: 20365275
## Max. :14.5400 Max. :16.0300 Max. :139958633
##
## MarketCap
## Min. :1.272e+07
## 1st Qu.:4.116e+07
## Median :2.400e+08
## Mean :2.860e+08
## 3rd Qu.:3.968e+08
## Max. :1.603e+09
## NA's :18waves %>% as.tibble %>% tail## # A tibble: 6 x 7
## Date Open High Low Close Volume MarketCap
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2019-08-30 1.12 1.14 1.11 1.12 7948837 111875703
## 2 2019-08-31 1.12 1.14 1.08 1.11 8863325 110914531
## 3 2019-09-01 1.11 1.12 1.08 1.12 7635752 111861896
## 4 2019-09-02 1.12 1.16 1.1 1.15 9763244 115401844
## 5 2019-09-03 1.15 1.17 1.13 1.15 11907801 115202292
## 6 2019-09-04 1.15 1.16 1.11 1.13 9476856 113087466nano <- scrapData("nano")
summary(nano)## Date Open High
## Min. :2017-03-07 Min. : 0.00729 Min. : 0.00795
## 1st Qu.:2017-11-01 1st Qu.: 0.14547 1st Qu.: 0.15388
## Median :2018-06-13 Median : 1.19000 Median : 1.26000
## Mean :2018-06-12 Mean : 2.86186 Mean : 3.09433
## 3rd Qu.:2019-01-23 3rd Qu.: 2.42000 3rd Qu.: 2.58000
## Max. :2019-09-04 Max. :33.45000 Max. :37.62000
## Low Close Volume
## Min. : 0.006658 Min. : 0.00729 Min. : 663
## 1st Qu.: 0.130532 1st Qu.: 0.14712 1st Qu.: 203958
## Median : 1.130000 Median : 1.19000 Median : 3746730
## Mean : 2.581955 Mean : 2.85865 Mean : 10874235
## 3rd Qu.: 2.290000 3rd Qu.: 2.42000 3rd Qu.: 11503800
## Max. :29.630000 Max. :33.70000 Max. :396790016
## MarketCap
## Min. :3.047e+05
## 1st Qu.:1.661e+07
## Median :1.586e+08
## Mean :3.804e+08
## 3rd Qu.:3.224e+08
## Max. :4.491e+09nano %>% as.tibble %>% tail## # A tibble: 6 x 7
## Date Open High Low Close Volume MarketCap
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2019-08-30 0.950 0.976 0.934 0.970 2748574 129266612
## 2 2019-08-31 0.970 0.976 0.935 0.958 2165914 127657898
## 3 2019-09-01 0.958 0.975 0.936 0.966 2162533 128736864
## 4 2019-09-02 0.966 1.02 0.958 1.01 2696596 135094914
## 5 2019-09-03 1.01 1.02 0.973 0.977 3499391 130168905
## 6 2019-09-04 0.977 0.983 0.958 0.970 2511865 129232255###################################################################################################
tron$Date<- as.Date(tron$Date)
tron <- tron[, c("Date", "Close")]
colnames(tron) <- c("Date", "tron_closure")
neo$Date<- as.Date(neo$Date)
neo <- neo[, c("Date", "Close")]
colnames(neo) <- c("Date", "neo_closure")
waves$Date<- as.Date(waves$Date)
waves <- waves[, c("Date", "Close")]
colnames(waves) <- c("Date", "waves_closure")
nano$Date<- as.Date(nano$Date)
nano <- nano[, c("Date", "Close")]
colnames(nano) <- c("Date", "nano_closure")
tron <- tron[tron$Date >= "2019-01-01", ]
neo <- neo[neo$Date >= "2019-01-01", ]
waves <- waves[waves$Date >= "2019-01-01", ]
nano <- nano[nano$Date >= "2019-01-01", ]
library(xts)
tron$rtron <- diff.xts(log(tron$tron))
neo$rneo <- diff.xts(log(neo$neo))
waves$rwaves <- diff.xts(log(waves$waves))
nano$rnano <- diff.xts(log(nano$nano))Następnym krokiem było utworzenie portfela oraz wstępna analiza zgromadzonych danych.
df <- data.frame(tron$Date, tron$rtron, neo$rneo, waves$rwaves, nano$rnano)
colnames(df) <- c("Date", "tron", "neo", "waves", "nano")
df$portfel <- rowSums(df[,c(2:5)])
portfel <- df[,c(1,6)]
colnames(portfel) <- c("Date", "zwroty")- Wykres gęstości zwrotów poszczególnych kryptowalut:
#install.packages("reshape",repos = "http://cran.us.r-project.org")
library(reshape)##
## Attaching package: 'reshape'## The following object is masked from 'package:dplyr':
##
## rename## The following objects are masked from 'package:tidyr':
##
## expand, smithsmolten.data <- melt(df,
id = c("Date"))
ggplot(molten.data,
aes(x = value, fill = variable)) +
geom_density(alpha = .3) +
xlab ("") + ylab("gęstość") +
ggtitle("Gęstości zwrotów poszczególnych kryptowalut")## Warning: Removed 5 rows containing non-finite values (stat_density).
2. Badanie niestacjonarność poszczególnych szeregów czasowych.
# tron, neo, waves, nano
par(mfrow = c(2, 1))
plot(tron$Date, tron$rtron,
type = "l", col="red", lwd = 1,
main = "zwroty TRON")
plot(tron$Date, tron$tron,
type = "l", col = "black", lwd = 1,
main = "notowania TRON")
par(mfrow = c(1, 1))
par(mfrow = c(2, 1))
plot(neo$Date, neo$rneo,
type = "l", col="red", lwd = 1,
main = "zwroty NEO")
plot(neo$Date, neo$neo,
type = "l", col = "black", lwd = 1,
main = "notowania NEO")
par(mfrow = c(1, 1))
par(mfrow = c(2, 1))
plot(waves$Date, waves$rwaves,
type = "l", col="red", lwd = 1,
main = "zwroty WAVES")
plot(waves$Date, waves$waves,
type = "l", col = "black", lwd = 1,
main = "notowania WAVES")
par(mfrow = c(1, 1))
par(mfrow = c(2, 1))
plot(nano$Date, nano$rnano,
type = "l", col="red", lwd = 1,
main = "zwroty NANO")
plot(nano$Date, nano$nano,
type = "l", col = "black", lwd = 1,
main = "notowania NANO")
par(mfrow = c(1, 1))Na podstawie wykresów trudno stwierdzić stacjonarność szeregów. Wydaje się, że średnia może mieć stałą wartość oczekiwaną, natomiast ciężko to powiedzieć o wariancji.
- Testowanie niestacjonarność poszczególnych szeregów czasowych.
testdf <- function(variable, adf_order) {
results_adf <- data.frame(order = -1,
adf = 0,
p_adf = "",
bgodfrey = 0, p_bg = 0)
variable <- variable[!is.na(variable)]
for (order in 0:adf_order) {
df.test_ <- ur.df(variable, type = c("drift"), lags = order)
df_ <- df.test_@teststat[1]
df_crit <- df.test_@cval[1, ]
df_crit <- (df_ < df_crit) * 1
p_adf <- ifelse(sum(df_crit) == 0,
">10pct",
paste("<",
names(df_crit)[min(which(df_crit == 1))],
sep = "")
)
resids_ <- df.test_@testreg$residuals
bgtest_ <- bgtest(resids_ ~ 1, order = 1)
bgodfrey <- bgtest_$statistic
names(bgodfrey) <- NULL
p_bg <- bgtest_$p.value
results_adf <- rbind(results_adf,
data.frame(order = order,
adf = df_,
p_adf = p_adf,
bgodfrey = bgodfrey,
p_bg = p_bg)
)
}
results_adf <- results_adf[results_adf$order >= 0, ]
plot(variable,
type = "l",
col = "blue",
lwd = 2,
main = "Plot of the examined variable")
return(results_adf)
}
# tron, neo, waves, nano
df.test <- ur.df(tron$tron, type = c("drift"), lags = 0)
summary(df.test)##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression drift
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0041952 -0.0005973 -0.0000843 0.0005849 0.0046388
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0008251 0.0004601 1.793 0.0742 .
## z.lag.1 -0.0326734 0.0175531 -1.861 0.0639 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.001353 on 244 degrees of freedom
## Multiple R-squared: 0.014, Adjusted R-squared: 0.00996
## F-statistic: 3.465 on 1 and 244 DF, p-value: 0.06389
##
##
## Value of test-statistic is: -1.8614 1.7499
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1 6.52 4.63 3.81df.test2 <- ur.df(neo$neo, type = c("drift"), lags = 0)
summary(df.test2)##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression drift
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.0792 -0.2702 0.0014 0.2529 3.5368
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.30167 0.16078 1.876 0.0618 .
## z.lag.1 -0.02769 0.01454 -1.904 0.0581 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6199 on 244 degrees of freedom
## Multiple R-squared: 0.01464, Adjusted R-squared: 0.0106
## F-statistic: 3.625 on 1 and 244 DF, p-value: 0.0581
##
##
## Value of test-statistic is: -1.9039 1.8203
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1 6.52 4.63 3.81df.test3 <- ur.df(waves$waves, type = c("drift"), lags = 0)
summary(df.test3)##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression drift
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.39195 -0.04980 -0.00308 0.04624 0.73916
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.01970 0.02681 0.735 0.463
## z.lag.1 -0.01218 0.01132 -1.076 0.283
##
## Residual standard error: 0.1039 on 244 degrees of freedom
## Multiple R-squared: 0.004722, Adjusted R-squared: 0.0006428
## F-statistic: 1.158 on 1 and 244 DF, p-value: 0.283
##
##
## Value of test-statistic is: -1.0759 1.3551
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1 6.52 4.63 3.81df.test4 <- ur.df(nano$nano, type = c("drift"), lags = 0)
summary(df.test4)##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression drift
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.17680 -0.03384 -0.00477 0.03155 0.29804
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.02983 0.01764 1.691 0.0921 .
## z.lag.1 -0.02445 0.01403 -1.743 0.0826 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06851 on 244 degrees of freedom
## Multiple R-squared: 0.01229, Adjusted R-squared: 0.008246
## F-statistic: 3.037 on 1 and 244 DF, p-value: 0.08265
##
##
## Value of test-statistic is: -1.7427 1.5185
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1 6.52 4.63 3.81P-value testu jest większe od 5% w każdym przypadku. Zatem brak jest podstaw do odrzucenia H0 o niestacjonarności szeregów.
Analiza portfela zrównoważonego
- Wykres zwrotów z portfela. ```
plot(portfel$Date, portfel$zwroty,
type = "l", col="blue", lwd = 1,
main = "zwroty z portfela")
- Testowanie normalności rozkładu zwrotów.
histogram zwrotów z portfela
library(fGarch)
bstats <- basicStats(portfel$zwroty)
knitr::kable(as.matrix(bstats), digits = 2)| X..portfel.zwroty | |
|---|---|
| nobs | 247.00 |
| NAs | 1.00 |
| Minimum | -0.53 |
| Maximum | 0.54 |
| 1. Quartile | -0.08 |
| 3. Quartile | 0.08 |
| Mean | 0.00 |
| Median | 0.00 |
| Sum | -1.10 |
| SE Mean | 0.01 |
| LCL Mean | -0.02 |
| UCL Mean | 0.02 |
| Variance | 0.03 |
| Stdev | 0.16 |
| Skewness | -0.22 |
| Kurtosis | 1.48 |
hist(portfel$zwroty, prob = T, breaks = 40)
curve(dnorm(x, mean = mean(portfel$zwroty, na.rm = T),
sd = sd(portfel$zwroty, na.rm = T)),
col = "blue", lwd = 2, add = TRUE)
# test jarque- bera
jarque.bera.test <-
function(x)
{
if((NCOL(x) > 1) || is.data.frame(x))
stop("x is not a vector or univariate time series")
if(any(is.na(x)))
stop("NAs in x")
DNAME <- deparse(substitute(x))
n <- length(x)
m1 <- sum(x)/n
m2 <- sum((x-m1)^2)/n
m3 <- sum((x-m1)^3)/n
m4 <- sum((x-m1)^4)/n
b1 <- (m3/m2^(3/2))^2
b2 <- (m4/m2^2)
STATISTIC <- n*b1/6+n*(b2-3)^2/24
PVAL <- 1 - pchisq(STATISTIC,df = 2)
PARAMETER <- 2
METHOD <- "Jarque Bera Test"
names(STATISTIC) <- "X-squared"
names(PARAMETER) <- "df"
structure(list(statistic = STATISTIC,
parameter = PARAMETER,
p.value = PVAL,
method = METHOD,
data.name = DNAME),
class = "htest")
}
jarque.bera.test(na.omit(portfel$zwroty))##
## Jarque Bera Test
##
## data: na.omit(portfel$zwroty)
## X-squared = 25.598, df = 2, p-value = 2.764e-06durbinWatsonTest(lm(portfel$zwroty ~ 1),
max.lag = 5)## lag Autocorrelation D-W Statistic p-value
## 1 -0.11575977 2.230080 0.066
## 2 -0.07875211 2.154012 0.192
## 3 0.02196341 1.949000 0.732
## 4 -0.03211117 2.056394 0.540
## 5 0.02745487 1.927502 0.788
## Alternative hypothesis: rho[lag] != 0Badanie normalności reszt - test Jarque Bera Badanie autokorelacji stóp zwrotów- statystyki Durbina-Watsona
Odrzucamy założenie o normalności reszt. P-value dla wszystkich opóźnień przekracza poziom istotności 5%, brak podstaw do odrzucenia H0 o braku autokorelacji zwrotów.
- Testowanie występowania efektów ARCH
ArchTest <- function (x, lags=12, demean = FALSE)
{
# Capture name of x for documentation in the output
xName <- deparse(substitute(x))
#
x <- as.vector(x)
if(demean) x <- scale(x, center = TRUE, scale = FALSE)
#
lags <- lags + 1
mat <- embed(x^2, lags)
arch.lm <- summary(lm(mat[, 1] ~ mat[, -1]))
STATISTIC <- arch.lm$r.squared * length(resid(arch.lm))
names(STATISTIC) <- "Chi-squared"
PARAMETER <- lags - 1
names(PARAMETER) <- "df"
PVAL <- 1 - pchisq(STATISTIC, df = PARAMETER)
METHOD <- "ARCH LM-test; Null hypothesis: no ARCH effects"
result <- list(statistic = STATISTIC, parameter = PARAMETER,
p.value = PVAL, method = METHOD, data.name =
xName)
class(result) <- "htest"
return(result)
}
ArchTest(portfel$zwroty, lags = 5)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: portfel$zwroty
## Chi-squared = 14.797, df = 5, p-value = 0.01126Odrzucamy H0 o braku efektów ARCH.
- Sprawdzenie, czy ACF reszt i ich kwadratów zachowują się jak biały szum.
Wykresy ACF dla zwrotów i kwadratów zwrotów.
acf(portfel$zwroty, lag.max = 36, na.action = na.pass,
col = "blue", lwd = 7,
main = "Wykres ACF zwrotów portfela")
acf(portfel$zwroty^2, lag.max = 100, na.action = na.pass,
col = "blue", lwd = 7,
main = "Wykres ACF kwadratów zwrotów portfela")
Na podstawie wykresów wnioskujemy, że nie zachowują się jak biały szum.
- Badanie autokorelacji zmiennej
Wykres wartości ACF dla zwrotów: 
Badana zmienna podlega autokorelacji.
Estymacja modelów GARCH w celu wyboru odpowiedniego modelu
Tworzymy próbkę ograniczoną do pewnego okresu czasu. Generujemy GARCH o różnych rzędach p i q:
portfel_probka_in <- portfel[portfel$Date <= as.Date("2019-05-01"), ]
portfel_probka_in <- portfel_probka_in[as.Date("2019-03-01") <= portfel_probka_in$Date, ]
library(fGarch)
spec_11 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_11 <- ugarchfit(spec = spec_11,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationspec_10 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1, 0)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_10 <- ugarchfit(spec = spec_10,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationspec_20 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(2, 0)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_20 <- ugarchfit(spec = spec_20,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationspec_21 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_21 <- ugarchfit(spec = spec_21,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationspec_30 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(3, 0)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_30 <- ugarchfit(spec = spec_30,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationspec_31 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(3, 1)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_31 <- ugarchfit(spec = spec_31,
data = na.omit(portfel_probka_in$zwroty))## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationgarch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.017410 0.448338 0.65391
## omega 0.000105 0.001974 0.053073 0.95767
## alpha1 0.000000 0.048064 0.000000 1.00000
## beta1 0.999000 0.127840 7.814467 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.011901 0.655892 0.511894
## omega 0.000105 0.003070 0.034119 0.972782
## alpha1 0.000000 0.114887 0.000000 1.000000
## beta1 0.999000 0.242399 4.121296 0.000038
##
## LogLikelihood : 37.53098
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.08164
## Bayes -0.94441
## Shibata -1.08932
## Hannan-Quinn -1.02776
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.332 0.2484
## Lag[2*(p+q)+(p+q)-1][2] 1.379 0.3901
## Lag[4*(p+q)+(p+q)-1][5] 2.567 0.4917
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01737 0.8951
## Lag[2*(p+q)+(p+q)-1][5] 0.75792 0.9116
## Lag[4*(p+q)+(p+q)-1][9] 2.60867 0.8214
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.07056 0.500 2.000 0.7905
## ARCH Lag[5] 0.45740 1.440 1.667 0.8961
## ARCH Lag[7] 1.19966 2.315 1.543 0.8791
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.6032
## Individual Statistics:
## mu 0.16164
## omega 0.05206
## alpha1 0.05538
## beta1 0.04945
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.2212 0.8257
## Negative Sign Bias 0.1040 0.9175
## Positive Sign Bias 0.1687 0.8666
## Joint Effect 0.0537 0.9967
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.77 0.7368
## 2 30 26.06 0.6220
## 3 40 27.03 0.9262
## 4 50 46.06 0.5929
##
##
## Elapsed time : 0.10078garch_10##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.022708 0.000035 655.396 0
## omega 0.000046 0.000000 12725.898 0
## alpha1 0.998988 0.018431 54.201 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.022708 0.000033 683.288 0
## omega 0.000046 0.000000 4755.550 0
## alpha1 0.998988 0.017693 56.461 0
##
## LogLikelihood : -171.9577
##
## Information Criteria
## ------------------------------------
##
## Akaike 5.6438
## Bayes 5.7467
## Shibata 5.6394
## Hannan-Quinn 5.6842
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01618 0.8988
## Lag[2*(p+q)+(p+q)-1][2] 0.04087 0.9635
## Lag[4*(p+q)+(p+q)-1][5] 0.65108 0.9321
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04859 0.8255
## Lag[2*(p+q)+(p+q)-1][2] 0.07430 0.9381
## Lag[4*(p+q)+(p+q)-1][5] 0.12826 0.9969
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 0.04821 0.500 2.000 0.8262
## ARCH Lag[4] 0.08131 1.397 1.611 0.9879
## ARCH Lag[6] 0.11194 2.222 1.500 0.9988
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.1168
## Individual Statistics:
## mu 0.1849
## omega 2.9172
## alpha1 3.2082
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.3815 0.1725
## Negative Sign Bias 1.6449 0.1055
## Positive Sign Bias 0.1268 0.8995
## Joint Effect 3.4274 0.3303
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 45.74 5.391e-04
## 2 30 64.77 1.518e-04
## 3 40 105.74 4.528e-08
## 4 50 141.23 7.130e-11
##
##
## Elapsed time : 0.09729886garch_20##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006944 0.017099 0.406135 0.684644
## omega 0.018196 0.003669 4.959760 0.000001
## alpha1 0.000000 0.100299 0.000002 0.999998
## alpha2 0.000000 0.082562 0.000000 1.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006944 0.015672 0.443123 0.657677
## omega 0.018196 0.005628 3.233144 0.001224
## alpha1 0.000000 0.054268 0.000004 0.999997
## alpha2 0.000000 0.044318 0.000000 1.000000
##
## LogLikelihood : 37.12038
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.06840
## Bayes -0.93116
## Shibata -1.07607
## Hannan-Quinn -1.01452
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.231 0.2673
## Lag[2*(p+q)+(p+q)-1][2] 1.276 0.4168
## Lag[4*(p+q)+(p+q)-1][5] 2.506 0.5045
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.007693 0.9301
## Lag[2*(p+q)+(p+q)-1][5] 0.720749 0.9189
## Lag[4*(p+q)+(p+q)-1][9] 2.565156 0.8280
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.07805 0.500 2.000 0.7800
## ARCH Lag[5] 0.38792 1.440 1.667 0.9161
## ARCH Lag[7] 1.14774 2.315 1.543 0.8884
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.4985
## Individual Statistics:
## mu 0.1769
## omega 0.1721
## alpha1 0.1867
## alpha2 0.7394
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.22974 0.8191
## Negative Sign Bias 0.16873 0.8666
## Positive Sign Bias 0.17575 0.8611
## Joint Effect 0.06553 0.9956
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 13.48 0.8129
## 2 30 28.97 0.4667
## 3 40 38.65 0.4859
## 4 50 47.68 0.5268
##
##
## Elapsed time : 0.138262garch_21##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007841 0.022847 0.343175 0.731467
## omega 0.000110 0.007705 0.014233 0.988644
## alpha1 0.000000 0.560631 0.000000 1.000000
## alpha2 0.000000 0.407393 0.000000 1.000000
## beta1 0.999000 0.515859 1.936575 0.052797
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007841 0.115623 0.067812 0.94593
## omega 0.000110 0.059377 0.001847 0.99853
## alpha1 0.000000 4.281730 0.000000 1.00000
## alpha2 0.000000 2.990288 0.000000 1.00000
## beta1 0.999000 4.022204 0.248371 0.80385
##
## LogLikelihood : 37.55217
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.05007
## Bayes -0.87853
## Shibata -1.06183
## Hannan-Quinn -0.98272
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.338 0.2473
## Lag[2*(p+q)+(p+q)-1][2] 1.385 0.3886
## Lag[4*(p+q)+(p+q)-1][5] 2.571 0.4909
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01796 0.8934
## Lag[2*(p+q)+(p+q)-1][8] 1.52685 0.9241
## Lag[4*(p+q)+(p+q)-1][14] 5.92899 0.6484
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2241 0.500 2.000 0.6360
## ARCH Lag[6] 0.9465 1.461 1.711 0.7632
## ARCH Lag[8] 1.6141 2.368 1.583 0.8180
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.2465
## Individual Statistics:
## mu 0.16080
## omega 0.04962
## alpha1 0.05764
## alpha2 0.05215
## beta1 0.04731
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.22079 0.8260
## Negative Sign Bias 0.10085 0.9200
## Positive Sign Bias 0.16849 0.8668
## Joint Effect 0.05345 0.9968
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.77 0.7368
## 2 30 26.06 0.6220
## 3 40 27.03 0.9262
## 4 50 46.06 0.5929
##
##
## Elapsed time : 0.07806802garch_30##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(3,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.005816 0.017220 0.337721 0.735574
## omega 0.018070 0.003934 4.593297 0.000004
## alpha1 0.000000 0.103416 0.000002 0.999999
## alpha2 0.000000 0.082420 0.000000 1.000000
## alpha3 0.022437 0.079692 0.281545 0.778293
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.005816 0.016204 0.358897 0.719672
## omega 0.018070 0.006183 2.922281 0.003475
## alpha1 0.000000 0.058071 0.000003 0.999998
## alpha2 0.000000 0.045824 0.000000 1.000000
## alpha3 0.022437 0.045603 0.492005 0.622716
##
## LogLikelihood : 37.18918
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.03836
## Bayes -0.86682
## Shibata -1.05012
## Hannan-Quinn -0.97101
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.224 0.2686
## Lag[2*(p+q)+(p+q)-1][2] 1.272 0.4179
## Lag[4*(p+q)+(p+q)-1][5] 2.494 0.5072
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.003594 0.9522
## Lag[2*(p+q)+(p+q)-1][8] 1.269796 0.9518
## Lag[4*(p+q)+(p+q)-1][14] 5.454479 0.7115
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.1775 0.500 2.000 0.6735
## ARCH Lag[6] 0.7457 1.461 1.711 0.8215
## ARCH Lag[8] 1.3605 2.368 1.583 0.8652
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.5763
## Individual Statistics:
## mu 0.17970
## omega 0.14824
## alpha1 0.17388
## alpha2 0.75575
## alpha3 0.08148
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.28404 0.7774
## Negative Sign Bias 0.23007 0.8189
## Positive Sign Bias 0.17455 0.8621
## Joint Effect 0.09603 0.9923
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.77 0.7368
## 2 30 24.13 0.7225
## 3 40 38.65 0.4859
## 4 50 42.84 0.7199
##
##
## Elapsed time : 0.167423garch_31##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(3,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007881 0.020894 0.37720 0.70603
## omega 0.000115 0.009396 0.01221 0.99026
## alpha1 0.000000 0.658353 0.00000 1.00000
## alpha2 0.000000 0.747821 0.00000 1.00000
## alpha3 0.000000 0.280928 0.00000 1.00000
## beta1 0.999000 0.622721 1.60425 0.10866
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007881 0.222268 0.035458 0.97171
## omega 0.000115 0.076655 0.001497 0.99881
## alpha1 0.000000 5.404559 0.000000 1.00000
## alpha2 0.000000 6.070504 0.000000 1.00000
## alpha3 0.000000 2.118972 0.000000 1.00000
## beta1 0.999000 5.036373 0.198357 0.84277
##
## LogLikelihood : 37.57291
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.01848
## Bayes -0.81263
## Shibata -1.03510
## Hannan-Quinn -0.93766
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.345 0.2462
## Lag[2*(p+q)+(p+q)-1][2] 1.392 0.3870
## Lag[4*(p+q)+(p+q)-1][5] 2.575 0.4900
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01856 0.8916
## Lag[2*(p+q)+(p+q)-1][11] 4.24114 0.6966
## Lag[4*(p+q)+(p+q)-1][19] 8.05072 0.6700
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.3576 0.500 2.000 0.5498
## ARCH Lag[7] 1.4420 1.473 1.746 0.6377
## ARCH Lag[9] 5.3536 2.402 1.619 0.2314
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.3685
## Individual Statistics:
## mu 0.15997
## omega 0.04751
## alpha1 0.05929
## alpha2 0.05413
## alpha3 0.04400
## beta1 0.04548
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.22029 0.8264
## Negative Sign Bias 0.09757 0.9226
## Positive Sign Bias 0.16823 0.8670
## Joint Effect 0.05320 0.9968
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 12.84 0.8467
## 2 30 23.16 0.7691
## 3 40 23.16 0.9793
## 4 50 42.84 0.7199
##
##
## Elapsed time : 0.07807183Dokonujemy porównania modeli uwzględniając następujące kryteria: AIC, BIC, Shibata oraz Hannan-Quinna.
``` -garch_31 Akaike -1.01848 Bayes -0.81263 Shibata -1.03510 Hannan-Quinn -0.93766
-garch_30 Akaike -1.03836 Bayes -0.86682 Shibata -1.05012 Hannan-Quinn -0.97101
-garch_21 Akaike -1.05007 Bayes -0.87853 Shibata -1.06183 Hannan-Quinn -0.98272
-garch_20 Akaike -1.06840 Bayes -0.93116 Shibata -1.07607 Hannan-Quinn -1.01452
-garch_10 Akaike 5.6438 Bayes 5.7467 Shibata 5.6394 Hannan-Quinn 5.6842
-garch_11 Akaike -1.08164 Bayes -0.94441 Shibata -1.08932 Hannan-Quinn -1.02776
Najniższe kryteria przyjmuje model GARCH(1,1).
garch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.017410 0.448338 0.65391
## omega 0.000105 0.001974 0.053073 0.95767
## alpha1 0.000000 0.048064 0.000000 1.00000
## beta1 0.999000 0.127840 7.814467 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.011901 0.655892 0.511894
## omega 0.000105 0.003070 0.034119 0.972782
## alpha1 0.000000 0.114887 0.000000 1.000000
## beta1 0.999000 0.242399 4.121296 0.000038
##
## LogLikelihood : 37.53098
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.08164
## Bayes -0.94441
## Shibata -1.08932
## Hannan-Quinn -1.02776
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.332 0.2484
## Lag[2*(p+q)+(p+q)-1][2] 1.379 0.3901
## Lag[4*(p+q)+(p+q)-1][5] 2.567 0.4917
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01737 0.8951
## Lag[2*(p+q)+(p+q)-1][5] 0.75792 0.9116
## Lag[4*(p+q)+(p+q)-1][9] 2.60867 0.8214
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.07056 0.500 2.000 0.7905
## ARCH Lag[5] 0.45740 1.440 1.667 0.8961
## ARCH Lag[7] 1.19966 2.315 1.543 0.8791
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.6032
## Individual Statistics:
## mu 0.16164
## omega 0.05206
## alpha1 0.05538
## beta1 0.04945
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.2212 0.8257
## Negative Sign Bias 0.1040 0.9175
## Positive Sign Bias 0.1687 0.8666
## Joint Effect 0.0537 0.9967
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.77 0.7368
## 2 30 26.06 0.6220
## 3 40 27.03 0.9262
## 4 50 46.06 0.5929
##
##
## Elapsed time : 0.10078Testy wskazują na to, że nie występuje autokorelacja reszt i kwadratów reszt wystandaryzowanych, ani efekty ARCH.
Następnie utworzono wykresy funkcji autokorelacji dla kwadratów reszt:
plot(garch_11, which = 10)
plot(garch_20, which = 10)
plot(garch_10, which = 10)
plot(garch_21, which = 10)
plot(garch_30, which = 10)
plot(garch_31, which = 10)
Wypustki są nieistotne, ponieważ nie przekraczają obszaru krytycznego. Zatem nie występuje autokorelacja reszt.
Kolejnym krokiem jest obliczenie Value-At_Risk w in sample i out of sample oraz porównanie kwantyli empirycznych ze standardowym z rozkładu normalnego.
portfel_probka_in$zwrotystd <- (portfel_probka_in$zwroty - mean(portfel_probka_in$zwroty, na.rm=T)) /
sd(portfel_probka_in$zwroty ,na.rm = T)
tail(portfel_probka_in)## Date zwroty zwrotystd
## 116 2019-04-26 -0.02431955 -0.2332355
## 117 2019-04-27 -0.01574112 -0.1692550
## 118 2019-04-28 -0.06799414 -0.5589739
## 119 2019-04-29 -0.16912923 -1.3132701
## 120 2019-04-30 0.29343999 2.1367114
## 121 2019-05-01 -0.06367050 -0.5267269basicStats(portfel_probka_in$zwrotystd)## X..portfel_probka_in.zwrotystd
## nobs 62.000000
## NAs 0.000000
## Minimum -2.831300
## Maximum 3.986365
## 1. Quartile -0.510065
## 3. Quartile 0.520829
## Mean 0.000000
## Median -0.059323
## Sum 0.000000
## SE Mean 0.127000
## LCL Mean -0.253952
## UCL Mean 0.253952
## Variance 1.000000
## Stdev 1.000000
## Skewness 0.632851
## Kurtosis 3.294645q01 <- quantile(portfel_probka_in$zwrotystd, 0.01, na.rm = T)
q01## 1%
## -2.297398qnorm(0.01, 0, 1)## [1] -2.326348str(garch_11)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:4, 1:4] 3593 -23556 -810 -522 -23556 ...
## .. ..$ cvar : num [1:4, 1:4] 3.03e-04 4.57e-06 2.26e-04 -3.87e-04 4.57e-06 ...
## .. ..$ var : num [1:62] 0.0177 0.0178 0.0179 0.0179 0.018 ...
## .. ..$ sigma : num [1:62] 0.133 0.133 0.134 0.134 0.134 ...
## .. ..$ condH : num 6.19
## .. ..$ z : num [1:62] 0.293 -0.437 -0.419 -0.685 1.064 ...
## .. ..$ LLH : num 37.5
## .. ..$ log.likelihoods: num [1:62] -1.056 -1.001 -1.006 -0.856 -0.522 ...
## .. ..$ residuals : num [1:62] 0.0389 -0.0582 -0.056 -0.0918 0.143 ...
## .. ..$ coef : Named num [1:4] 7.81e-03 1.05e-04 3.05e-09 9.99e-01
## .. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. ..$ robust.cvar : num [1:4, 1:4] 1.42e-04 9.19e-06 4.07e-04 -7.66e-04 9.19e-06 ...
## .. ..$ A : num [1:4, 1:4] 57.95 -379.93 -13.07 -8.42 -379.93 ...
## .. ..$ B : num [1:4, 1:4] 37.2 3032.2 12.8 57.3 3032.2 ...
## .. ..$ scores : num [1:62, 1:4] -2.16 3.31 3.18 5.14 -7.93 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. ..$ se.coef : num [1:4] 0.01741 0.00197 0.04806 0.12784
## .. ..$ tval : Named num [1:4] 4.48e-01 5.31e-02 6.34e-08 7.81
## .. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. ..$ matcoef : num [1:4, 1:4] 7.81e-03 1.05e-04 3.05e-09 9.99e-01 1.74e-02 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:4] 0.0119 0.00307 0.11489 0.2424
## .. ..$ robust.tval : Named num [1:4] 6.56e-01 3.41e-02 2.65e-08 4.12
## .. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. ..$ robust.matcoef : num [1:4, 1:4] 7.81e-03 1.05e-04 3.05e-09 9.99e-01 1.19e-02 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:62] 0.00781 0.00781 0.00781 0.00781 0.00781 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 1
## .. ..$ persistence : num 0.999
## .. ..$ timer : 'difftime' num 0.100780010223389
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:19, 1:6] 0.00781 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:4] 7.81e-03 1.05e-04 3.05e-09 9.99e-01
## .. .. .. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:3] -10.5 -37.5 -37.5
## .. .. .. ..$ lagrange : num [1, 1] -3.24
## .. .. .. ..$ hessian : num [1:5, 1:5] 0.3338 0.0288 -7.6954 -1.1607 -0.7206 ...
## .. .. .. ..$ ineqx0 : Named num 0.999
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 185
## .. .. .. ..$ outer.iter : num 2
## .. .. .. ..$ elapsed : 'difftime' num 0.0850160121917725
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:6] 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 1 0 0 0 0 0 1 1 1 0 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "sGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:19, 1:6] 0.00781 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 1 0 0 0 0 0 2 3 4 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0garch_11@fit$sigma## [1] 0.1329958 0.1333227 0.1336486 0.1339732 0.1342968 0.1346193 0.1349407
## [8] 0.1352610 0.1355802 0.1358984 0.1362155 0.1365315 0.1368465 0.1371605
## [15] 0.1374734 0.1377854 0.1380962 0.1384061 0.1387150 0.1390229 0.1393298
## [22] 0.1396358 0.1399407 0.1402447 0.1405477 0.1408498 0.1411509 0.1414511
## [29] 0.1417503 0.1420487 0.1423460 0.1426425 0.1429381 0.1432328 0.1435265
## [36] 0.1438194 0.1441114 0.1444025 0.1446927 0.1449821 0.1452706 0.1455582
## [43] 0.1458450 0.1461309 0.1464160 0.1467002 0.1469836 0.1472662 0.1475480
## [50] 0.1478289 0.1481091 0.1483884 0.1486669 0.1489446 0.1492216 0.1494977
## [57] 0.1497731 0.1500476 0.1503214 0.1505945 0.1508667 0.1511382portfel_probka_in$VaR <- q01 * garch_11@fit$sigma
tail(portfel_probka_in)## Date zwroty zwrotystd VaR
## 116 2019-04-26 -0.02431955 -0.2332355 -0.3440883
## 117 2019-04-27 -0.01574112 -0.1692550 -0.3447191
## 118 2019-04-28 -0.06799414 -0.5589739 -0.3453481
## 119 2019-04-29 -0.16912923 -1.3132701 -0.3459754
## 120 2019-04-30 0.29343999 2.1367114 -0.3466009
## 121 2019-05-01 -0.06367050 -0.5267269 -0.3472246Kwantyle nie równią się od siebie w znaczącym stopniu.
Wykres zwrotów i wartości Value-At_Risk przedstawia się w następujący sposób:
plot(portfel_probka_in$Date, portfel_probka_in$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-2,2))
abline(h = 0, lty = 2)
lines(portfel_probka_in$Date, portfel_probka_in$VaR, type = 'l', col = "green")
sum(portfel_probka_in$zwroty < portfel_probka_in$VaR) / length(portfel_probka_in$VaR)## [1] 0.01612903Straty przekroczyły zakładany poziom VaR w 1,6% przypadków.
Natomiast dla out-of-sample:
#1-dniowa prognoza warunkowego odchylenia standardowego
sigma.forecast1 <- ugarchforecast(garch_11, n.ahead = 1)
str(sigma.forecast1)## Formal class 'uGARCHforecast' [package "rugarch"] with 2 slots
## ..@ forecast:List of 6
## .. ..$ n.ahead : num 1
## .. ..$ N : num 62
## .. ..$ n.start : num 0
## .. ..$ n.roll : num 0
## .. ..$ sigmaFor : num [1, 1] 0.151
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## .. ..$ seriesFor: num [1, 1] 0.00781
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## ..@ model :List of 11
## .. ..$ modelinc : Named num [1:22] 1 0 0 0 0 0 1 1 1 0 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "sGARCH"
## .. ..$ modeldata :List of 6
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period : 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. .. ..$ sigma : num [1:62] 0.133 0.133 0.134 0.134 0.134 ...
## .. .. ..$ residuals: num [1:62] 0.0389 -0.0582 -0.056 -0.0918 0.143 ...
## .. ..$ pars : num [1:19, 1:6] 0.00781 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 1 0 0 0 0 0 2 3 4 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0str(sigma.forecast1@forecast)## List of 6
## $ n.ahead : num 1
## $ N : num 62
## $ n.start : num 0
## $ n.roll : num 0
## $ sigmaFor : num [1, 1] 0.151
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"
## $ seriesFor: num [1, 1] 0.00781
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"sigma.forecast1@forecast$sigmaFor## 1970-03-04 01:00:00
## T+1 0.151409sigma.forecast1_2 <- sigma.forecast1@forecast$sigmaFor[1, 1]
# 1-dniowy VaR:
q01 * sigma.forecast1_2## 1%
## -0.3478466portfel$obs<-1:length(portfel$zwroty)
start <- portfel$obs[portfel$Date == as.Date("2019-03-02")]
finish <- portfel$obs[portfel$Date == as.Date("2019-06-01")]
portfel1 <-portfel[start:finish, ]
VaR1 <- rep(NA, times = finish - start + 1)
time1 <- Sys.time()
for (k in start:finish) {
tmp.data <- portfel[portfel$obs <= (k - 1), ]
tmp.data <- tmp.data[as.Date("2019-01-02") <= tmp.data$Date, ]
tmp.data$zwrotystd <- (tmp.data$zwroty - mean(tmp.data$zwroty, na.rm = T)) /
sd(tmp.data$zwroty, na.rm = T)
q01 <- quantile(tmp.data$zwrotystd, 0.01, na.rm = T)
spec10 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1, 0)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
tmp.garch10 <- ugarchfit(spec = spec_10, data = na.omit(portfel$zwroty))
sigma.forecast1 <- ugarchforecast(tmp.garch10, n.ahead = 1)
sigma.forecast12 <- sigma.forecast1@forecast$sigmaFor[1, 1]
VaR1[k - start + 1] <- q01 * sigma.forecast12
}
time2 <- Sys.time()
time2 - time1## Time difference of 12.23209 secsportfel1$VaR1 <- VaR1
plot(portfel1$Date, portfel1$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-1, 1))
abline(h = 0, lty = 2)
lines(portfel1$Date, portfel1$VaR1, type = 'l', col = "green")
sum(portfel1$zwroty < portfel1$VaR1) / length(portfel1$VaR1)## [1] 0Straty przekroczyły zakładany poziom VaR w 0% przypadków.
Szacowanie modelu tGARCH:
tspec_10 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(1, 0),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
k.tgarch_10 <- ugarchfit(spec = tspec_10, data = na.omit(portfel_probka_in$zwroty))## Warning in .fgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationtspec_11 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(1, 1),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
k.tgarch_11 <- ugarchfit(spec = tspec_11, data = na.omit(portfel_probka_in$zwroty))## Warning in .fgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationtspec_21 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(2, 1),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
k.tgarch_21 <- ugarchfit(spec = tspec_21, data = na.omit(portfel_probka_in$zwroty))## Warning in .fgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationtspec_30 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(3, 0),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
k.tgarch_30 <- ugarchfit(spec = tspec_30, data = na.omit(portfel_probka_in$zwroty))## Warning in .fgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationk.tgarch_10##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,0)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000018 0.000000 302.46 0
## alpha1 0.137546 0.000454 302.88 0
## eta11 0.653886 0.002187 298.99 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000018 0.000143 0.128586 0.89769
## alpha1 0.137546 1.946785 0.070653 0.94367
## eta11 0.653886 0.914960 0.714660 0.47482
##
## LogLikelihood : -3202.228
##
## Information Criteria
## ------------------------------------
##
## Akaike 103.39
## Bayes 103.50
## Shibata 103.39
## Hannan-Quinn 103.43
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.05409 0.8161
## Lag[2*(p+q)+(p+q)-1][2] 0.06007 0.9486
## Lag[4*(p+q)+(p+q)-1][5] 0.12687 0.9970
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.02870 0.8655
## Lag[2*(p+q)+(p+q)-1][2] 0.04130 0.9631
## Lag[4*(p+q)+(p+q)-1][5] 0.08264 0.9987
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 0.02362 0.500 2.000 0.8779
## ARCH Lag[4] 0.05440 1.397 1.611 0.9930
## ARCH Lag[6] 0.08574 2.222 1.500 0.9993
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.65
## Individual Statistics:
## omega 3.1707
## alpha1 0.2741
## eta11 0.2741
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.325 0.1903
## Negative Sign Bias 0.143 0.8868
## Positive Sign Bias 1.457 0.1505
## Joint Effect 3.030 0.3870
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 412.8 9.336e-76
## 2 30 649.3 3.693e-118
## 3 40 885.1 2.136e-160
## 4 50 1083.2 9.005e-195
##
##
## Elapsed time : 0.2447441k.tgarch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002279 0.004699 0.485102 0.6276
## alpha1 0.000000 0.000002 0.000131 0.9999
## beta1 0.991708 0.038038 26.071406 0.0000
## eta11 -0.985951 1.677084 -0.587896 0.5566
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002279 0.004043 0.56384 0.57286
## alpha1 0.000000 0.000000 0.30145 0.76307
## beta1 0.991708 0.029365 33.77200 0.00000
## eta11 -0.985951 4.316444 -0.22842 0.81932
##
## LogLikelihood : 39.61836
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.1490
## Bayes -1.0117
## Shibata -1.1567
## Hannan-Quinn -1.0951
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.806 0.1790
## Lag[2*(p+q)+(p+q)-1][2] 1.865 0.2862
## Lag[4*(p+q)+(p+q)-1][5] 2.986 0.4097
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06796 0.7943
## Lag[2*(p+q)+(p+q)-1][5] 0.86983 0.8884
## Lag[4*(p+q)+(p+q)-1][9] 2.68274 0.8100
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.08566 0.500 2.000 0.7698
## ARCH Lag[5] 0.70240 1.440 1.667 0.8227
## ARCH Lag[7] 1.50849 2.315 1.543 0.8197
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.2742
## Individual Statistics:
## omega 0.05682
## alpha1 0.05023
## beta1 0.05659
## eta11 0.06104
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.4090 0.6841
## Negative Sign Bias 0.1023 0.9189
## Positive Sign Bias 0.3188 0.7511
## Joint Effect 0.1904 0.9791
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.29 0.4384
## 2 30 23.16 0.7691
## 3 40 33.48 0.7191
## 4 50 34.77 0.9378
##
##
## Elapsed time : 0.1547849k.tgarch_21##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(2,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002704 0.005327 0.507608 0.61173
## alpha1 0.000000 0.000004 0.000823 0.99934
## alpha2 0.000000 0.000002 0.003638 0.99710
## beta1 0.988632 0.042721 23.141649 0.00000
## eta11 0.438115 0.571997 0.765940 0.44371
## eta12 0.999697 6.248077 0.160001 0.87288
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002704 0.004923 0.54930 0.58280
## alpha1 0.000000 0.000000 0.40615 0.68463
## alpha2 0.000000 0.000000 0.36311 0.71652
## beta1 0.988632 0.035885 27.55031 0.00000
## eta11 0.438115 1.646657 0.26606 0.79019
## eta12 0.999697 3.530842 0.28313 0.77707
##
## LogLikelihood : 39.67585
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.08632
## Bayes -0.88047
## Shibata -1.10294
## Hannan-Quinn -1.00550
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.829 0.1762
## Lag[2*(p+q)+(p+q)-1][2] 1.889 0.2818
## Lag[4*(p+q)+(p+q)-1][5] 3.014 0.4045
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06975 0.7917
## Lag[2*(p+q)+(p+q)-1][8] 1.79190 0.8896
## Lag[4*(p+q)+(p+q)-1][14] 5.50283 0.7051
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.3762 0.500 2.000 0.5396
## ARCH Lag[6] 1.3180 1.461 1.711 0.6588
## ARCH Lag[8] 2.0300 2.368 1.583 0.7355
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.2874
## Individual Statistics:
## omega 0.05579
## alpha1 0.05307
## alpha2 0.05399
## beta1 0.05563
## eta11 0.05946
## eta12 0.05648
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.4061 0.6862
## Negative Sign Bias 0.1079 0.9144
## Positive Sign Bias 0.3221 0.7486
## Joint Effect 0.1879 0.9795
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 17.35 0.5658
## 2 30 20.26 0.8847
## 3 40 29.61 0.8611
## 4 50 33.16 0.9595
##
##
## Elapsed time : 0.3316138k.tgarch_30##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(3,0)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.095976 0.004304 22.30122 0.00000
## alpha1 0.098090 0.111408 0.88045 0.37861
## alpha2 0.000000 0.000001 0.00000 1.00000
## alpha3 1.000000 0.074849 13.36019 0.00000
## eta11 0.972203 0.905793 1.07332 0.28313
## eta12 1.000000 1.129841 0.88508 0.37611
## eta13 -0.798866 0.040515 -19.71763 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.095976 0.006907 13.895990 0.000000
## alpha1 0.098090 0.145291 0.675124 0.499597
## alpha2 0.000000 0.000000 0.000308 0.999754
## alpha3 1.000000 0.366142 2.731182 0.006311
## eta11 0.972203 1.448094 0.671367 0.501986
## eta12 1.000000 3.706532 0.269794 0.787319
## eta13 -0.798866 0.062346 -12.813347 0.000000
##
## LogLikelihood : 36.86164
##
## Information Criteria
## ------------------------------------
##
## Akaike -0.96328
## Bayes -0.72312
## Shibata -0.98549
## Hannan-Quinn -0.86899
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.9146 0.3389
## Lag[2*(p+q)+(p+q)-1][2] 0.9816 0.5043
## Lag[4*(p+q)+(p+q)-1][5] 2.4036 0.5266
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.003778 0.9510
## Lag[2*(p+q)+(p+q)-1][8] 1.195486 0.9587
## Lag[4*(p+q)+(p+q)-1][14] 6.768152 0.5360
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.0427 0.500 2.000 0.8363
## ARCH Lag[6] 0.3540 1.461 1.711 0.9315
## ARCH Lag[8] 0.8131 2.368 1.583 0.9498
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.2707
## Individual Statistics:
## omega 0.10141
## alpha1 0.07344
## alpha2 0.36251
## alpha3 0.08342
## eta11 0.12555
## eta12 NaN
## eta13 0.06335
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.01261 0.9900
## Negative Sign Bias 0.17346 0.8629
## Positive Sign Bias 0.04569 0.9637
## Joint Effect 0.06357 0.9958
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.94 0.3985
## 2 30 26.06 0.6220
## 3 40 32.19 0.7716
## 4 50 55.74 0.2361
##
##
## Elapsed time : 1.08135k.tgarch_10 Akaike 103.39 Bayes 103.50 Shibata 103.39 Hannan-Quinn 103.43
k.tgarch_11 Akaike -1.1490 Bayes -1.0117 Shibata -1.1567 Hannan-Quinn -1.0951
k.tgarch_21 Akaike -1.08632 Bayes -0.88047 Shibata -1.10294 Hannan-Quinn -1.00550
k.tgarch_30 Akaike -0.96328 Bayes -0.72312 Shibata -0.98549 Hannan-Quinn -0.86899
Na podstawie kryteriów informacyjnych wybieramy model tGARCH(1,1).
Wykres funkcji autokorelacji dla tGARCH(1,1):
plot(k.tgarch_11, which = 10)
Wypustki są nieistotne, ponieważ nie przekraczają obszaru krytycznego. Zatem nie występuje autokorelacja reszt.
Kolejnym krokiem jest obliczenie Value-At_Risk w in sample i out of sample oraz porównanie kwantyli empirycznych ze standardowym z rozkładu normalnego.
portfel_probka_in$zwrotystd2 <- (portfel_probka_in$zwroty - mean(portfel_probka_in$zwroty, na.rm=T)) /
sd(portfel_probka_in$zwroty ,na.rm = T)
tail(portfel_probka_in)## Date zwroty zwrotystd VaR zwrotystd2
## 116 2019-04-26 -0.02431955 -0.2332355 -0.3440883 -0.2332355
## 117 2019-04-27 -0.01574112 -0.1692550 -0.3447191 -0.1692550
## 118 2019-04-28 -0.06799414 -0.5589739 -0.3453481 -0.5589739
## 119 2019-04-29 -0.16912923 -1.3132701 -0.3459754 -1.3132701
## 120 2019-04-30 0.29343999 2.1367114 -0.3466009 2.1367114
## 121 2019-05-01 -0.06367050 -0.5267269 -0.3472246 -0.5267269basicStats(portfel_probka_in$zwrotystd2)## X..portfel_probka_in.zwrotystd2
## nobs 62.000000
## NAs 0.000000
## Minimum -2.831300
## Maximum 3.986365
## 1. Quartile -0.510065
## 3. Quartile 0.520829
## Mean 0.000000
## Median -0.059323
## Sum 0.000000
## SE Mean 0.127000
## LCL Mean -0.253952
## UCL Mean 0.253952
## Variance 1.000000
## Stdev 1.000000
## Skewness 0.632851
## Kurtosis 3.294645q01 <- quantile(portfel_probka_in$zwrotystd2, 0.01, na.rm = T)
q01## 1%
## -2.297398qnorm(0.01, 0, 1)## [1] -2.326348str(k.tgarch_11)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:4, 1:4] 4.99e+06 4.90e+05 6.14e+05 -2.38e-05 4.90e+05 ...
## .. ..$ cvar : num [1:4, 1:4] 2.21e-05 -8.12e-13 -1.78e-04 6.68e-03 -8.12e-13 ...
## .. ..$ var : num [1:62] 0.00904 0.00932 0.00961 0.0099 0.01019 ...
## .. ..$ sigma : num [1:62] 0.0951 0.0966 0.098 0.0995 0.101 ...
## .. ..$ condH : num NaN
## .. ..$ z : num [1:62] 0.491 -0.522 -0.492 -0.844 1.493 ...
## .. ..$ LLH : num 39.6
## .. ..$ log.likelihoods: num [1:62] -1.314 -1.282 -1.282 -1.032 -0.259 ...
## .. ..$ residuals : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. ..$ coef : Named num [1:4] 2.28e-03 2.80e-10 9.92e-01 -9.86e-01
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. ..$ robust.cvar : num [1:4, 1:4] 1.63e-05 3.18e-14 -1.18e-04 -6.03e-03 3.18e-14 ...
## .. ..$ A : num [1:4, 1:4] 8.06e+04 7.90e+03 9.91e+03 -3.84e-07 7.90e+03 ...
## .. ..$ B : num [1:4, 1:4] 1.62e+05 1.49e+04 1.92e+04 -7.66e-07 1.49e+04 ...
## .. ..$ scores : num [1:62, 1:4] 0 7.53 15.4 8.58 -48.13 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. ..$ se.coef : num [1:4] 4.70e-03 2.14e-06 3.80e-02 1.68
## .. ..$ tval : Named num [1:4] 0.485102 0.000131 26.071406 -0.587896
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. ..$ matcoef : num [1:4, 1:4] 2.28e-03 2.80e-10 9.92e-01 -9.86e-01 4.70e-03 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:4] 4.04e-03 9.30e-10 2.94e-02 4.32
## .. ..$ robust.tval : Named num [1:4] 0.564 0.301 33.772 -0.228
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. ..$ robust.matcoef : num [1:4, 1:4] 2.28e-03 2.80e-10 9.92e-01 -9.86e-01 4.04e-03 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:62] 0 0 0 0 0 0 0 0 0 0 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 0.798
## .. ..$ persistence : num 0.992
## .. ..$ timer : 'difftime' num 0.154784917831421
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:4] 2.28e-03 2.80e-10 9.92e-01 -9.86e-01
## .. .. .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "eta11"
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:3] 148.2 -39.6 -39.6
## .. .. .. ..$ lagrange : num [1, 1] -5.4e-07
## .. .. .. ..$ hessian : num [1:5, 1:5] 0.461 -73.11 -7.171 -8.486 0.134 ...
## .. .. .. ..$ ineqx0 : Named num 0.992
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 166
## .. .. .. ..$ outer.iter : num 2
## .. .. .. ..$ elapsed : 'difftime' num 0.152949094772339
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:6] 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 0 0 0 0 0 0 1 1 1 0 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 4
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "fGARCH"
## .. .. ..$ vsubmodel : chr "TGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 0 0 0 0 0 0 1 2 3 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel :List of 4
## .. .. ..$ fpars :List of 5
## .. .. .. ..$ lambda: num 1
## .. .. .. ..$ delta : num 1
## .. .. .. ..$ eta1 : num 0.05
## .. .. .. ..$ eta2 : num 0
## .. .. .. ..$ fk : num 0
## .. .. ..$ finclude : num [1:5] 0 0 1 0 0
## .. .. ..$ fgarchtype: num 2
## .. .. ..$ fbounds :List of 2
## .. .. .. ..$ LB: num [1:5] 1 1 -1 0 0
## .. .. .. ..$ UB: num [1:5] 1 1 1 0 0
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0k.tgarch_11@fit$sigma## [1] 0.09506790 0.09655897 0.09803768 0.09950412 0.10095841 0.10240063
## [7] 0.10383090 0.10524931 0.10665596 0.10805094 0.10943436 0.11080631
## [13] 0.11216688 0.11351617 0.11485427 0.11618128 0.11749729 0.11880238
## [19] 0.12009665 0.12138019 0.12265309 0.12391543 0.12516731 0.12640880
## [25] 0.12764001 0.12886100 0.13007187 0.13127270 0.13246357 0.13364457
## [31] 0.13481578 0.13597727 0.13712913 0.13827145 0.13940429 0.14052774
## [37] 0.14164187 0.14274677 0.14384250 0.14492915 0.14600679 0.14707549
## [43] 0.14813534 0.14918639 0.15022873 0.15126243 0.15228755 0.15330418
## [49] 0.15431237 0.15531221 0.15630376 0.15728708 0.15826226 0.15922934
## [55] 0.16018841 0.16113952 0.16208275 0.16301816 0.16394581 0.16486577
## [61] 0.16577811 0.16668287portfel_probka_in$VaR2 <- q01 * k.tgarch_11@fit$sigma
tail(portfel_probka_in)## Date zwroty zwrotystd VaR zwrotystd2 VaR2
## 116 2019-04-26 -0.02431955 -0.2332355 -0.3440883 -0.2332355 -0.3723686
## 117 2019-04-27 -0.01574112 -0.1692550 -0.3447191 -0.1692550 -0.3745176
## 118 2019-04-28 -0.06799414 -0.5589739 -0.3453481 -0.5589739 -0.3766487
## 119 2019-04-29 -0.16912923 -1.3132701 -0.3459754 -1.3132701 -0.3787623
## 120 2019-04-30 0.29343999 2.1367114 -0.3466009 2.1367114 -0.3808582
## 121 2019-05-01 -0.06367050 -0.5267269 -0.3472246 -0.5267269 -0.3829369Wykres zwrotów i VaR.
plot(portfel_probka_in$Date, portfel_probka_in$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-2,2))
abline(h = 0, lty = 2)
lines(portfel_probka_in$Date, portfel_probka_in$VaR2, type = 'l', col = "green")
sum(portfel_probka_in$zwroty < portfel_probka_in$VaR2) / length(portfel_probka_in$VaR2)## [1] 0.01612903W 1,6% przypadkach straty przekroczyły zakładany poziom VaR.
Natomiast dla out of sample:
sigma.forecast2 <- ugarchforecast(k.tgarch_11, n.ahead = 1)
str(sigma.forecast2)## Formal class 'uGARCHforecast' [package "rugarch"] with 2 slots
## ..@ forecast:List of 6
## .. ..$ n.ahead : num 1
## .. ..$ N : num 62
## .. ..$ n.start : num 0
## .. ..$ n.roll : num 0
## .. ..$ sigmaFor : num [1, 1] 0.168
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## .. ..$ seriesFor: num [1, 1] 0
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## ..@ model :List of 11
## .. ..$ modelinc : Named num [1:22] 0 0 0 0 0 0 1 1 1 0 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 4
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "fGARCH"
## .. .. ..$ vsubmodel : chr "TGARCH"
## .. ..$ modeldata :List of 6
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period : 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. .. ..$ sigma : num [1:62] 0.0951 0.0966 0.098 0.0995 0.101 ...
## .. .. ..$ residuals: num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. ..$ pars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 0 0 0 0 0 0 1 2 3 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel :List of 4
## .. .. ..$ fpars :List of 5
## .. .. .. ..$ lambda: num 1
## .. .. .. ..$ delta : num 1
## .. .. .. ..$ eta1 : num 0.05
## .. .. .. ..$ eta2 : num 0
## .. .. .. ..$ fk : num 0
## .. .. ..$ finclude : num [1:5] 0 0 1 0 0
## .. .. ..$ fgarchtype: num 2
## .. .. ..$ fbounds :List of 2
## .. .. .. ..$ LB: num [1:5] 1 1 -1 0 0
## .. .. .. ..$ UB: num [1:5] 1 1 1 0 0
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0str(sigma.forecast2@forecast)## List of 6
## $ n.ahead : num 1
## $ N : num 62
## $ n.start : num 0
## $ n.roll : num 0
## $ sigmaFor : num [1, 1] 0.168
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"
## $ seriesFor: num [1, 1] 0
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"sigma.forecast2@forecast$sigmaFor## 1970-03-04 01:00:00
## T+1 0.1675801sigma.forecast22 <- sigma.forecast2@forecast$sigmaFor[1, 1]
# 1-dniowy VaR
q01 * sigma.forecast22## 1%
## -0.3849982portfel$obs<-1:length(portfel$zwroty)
start <- portfel$obs[portfel$Date == as.Date("2019-03-02")]
finish <- portfel$obs[portfel$Date == as.Date("2019-06-01")]
portfel4 <-portfel[start:finish, ]
VaR <- rep(NA, times = finish - start + 1)
time1 <- Sys.time()
for (k in start:finish) {
tmp.data <- portfel[portfel$obs <= (k - 1), ]
tmp.data <- tmp.data[as.Date("2019-01-02") <= tmp.data$Date, ]
tmp.data$zwrotystd <- (tmp.data$zwroty - mean(tmp.data$zwroty, na.rm = T)) /
sd(tmp.data$zwroty, na.rm = T)
q01 <- quantile(tmp.data$zwrotystd, 0.01, na.rm = T)
tspec11 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(2, 1),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
tmp.k.tgarch11 <- ugarchfit(spec = tspec_11, data = na.omit(portfel$zwroty))
sigma.forecast2 <- ugarchforecast(tmp.k.tgarch11, n.ahead = 1)
sigma.forecast22 <- sigma.forecast2@forecast$sigmaFor[1, 1]
VaR[k - start + 1] <- q01 * sigma.forecast22
}
time2 <- Sys.time()
time2 - time1## Time difference of 21.06684 secsportfel4$VaR <- VaR
plot(portfel4$Date, portfel4$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-1, 1))
abline(h = 0, lty = 2)
lines(portfel4$Date, portfel4$VaR, type = 'l', col = "green")
Straty przekroczyły zakładany poziom VaR w 0% przypadków.
Prognozy warunkowej wariancji są do siebie zbliżone.
Kolejnym krokiem było oszacowanie modelu eGARCH.
espec_11 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_11 <- ugarchfit(spec = espec_11, data = na.omit(portfel_probka_in$zwroty))## Warning in .egarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationespec_12 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_12 <- ugarchfit(spec = espec_12, data = na.omit(portfel_probka_in$zwroty))## Warning in .egarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationespec_21 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_21 <- ugarchfit(spec = espec_21, data = na.omit(portfel_probka_in$zwroty))## Warning in .egarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationespec_13 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(1, 3)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_13 <- ugarchfit(spec = espec_13, data = na.omit(portfel_probka_in$zwroty))## Warning in .egarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationespec_31 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(3, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_31 <- ugarchfit(spec = espec_31, data = na.omit(portfel_probka_in$zwroty))## Warning in .egarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationegarch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.202778 0.000014 -14626.742 0
## alpha1 0.011072 0.000155 71.452 0
## beta1 0.989998 0.000264 3743.420 0
## gamma1 -0.604025 0.000245 -2463.039 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.202778 0.000032 -6401.9881 0.000000
## alpha1 0.011072 0.004157 2.6633 0.007738
## beta1 0.989998 0.006251 158.3690 0.000000
## gamma1 -0.604025 0.001387 -435.4009 0.000000
##
## LogLikelihood : 39.83726
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.1560
## Bayes -1.0188
## Shibata -1.1637
## Hannan-Quinn -1.1022
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 4.157 0.04146
## Lag[2*(p+q)+(p+q)-1][2] 4.295 0.06283
## Lag[4*(p+q)+(p+q)-1][5] 5.763 0.10224
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.03917 0.8431
## Lag[2*(p+q)+(p+q)-1][5] 1.31294 0.7857
## Lag[4*(p+q)+(p+q)-1][9] 3.24404 0.7181
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.0005806 0.500 2.000 0.9808
## ARCH Lag[5] 0.2954018 1.440 1.667 0.9415
## ARCH Lag[7] 1.5187698 2.315 1.543 0.8176
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.7621
## Individual Statistics:
## omega 0.08437
## alpha1 0.08358
## beta1 0.01735
## gamma1 0.08428
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.3534 0.7251
## Negative Sign Bias 1.3397 0.1856
## Positive Sign Bias 0.3246 0.7466
## Joint Effect 2.3919 0.4951
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 27.03 0.1039
## 2 30 27.03 0.5700
## 3 40 37.35 0.5450
## 4 50 51.28 0.3845
##
##
## Elapsed time : 0.200274egarch_12##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.460382 0.000057 -8078.5 0
## alpha1 0.359605 0.000116 3112.3 0
## beta1 0.998011 0.000189 5273.1 0
## beta2 -0.063646 0.000013 -4890.2 0
## gamma1 -0.840378 0.000078 -10843.4 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.460382 0.004600 -100.077 0
## alpha1 0.359605 0.005514 65.223 0
## beta1 0.998011 0.013682 72.942 0
## beta2 -0.063646 0.000532 -119.545 0
## gamma1 -0.840378 0.008279 -101.502 0
##
## LogLikelihood : 42.884
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.2221
## Bayes -1.0505
## Shibata -1.2338
## Hannan-Quinn -1.1547
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.107 0.01346
## Lag[2*(p+q)+(p+q)-1][2] 6.404 0.01717
## Lag[4*(p+q)+(p+q)-1][5] 7.505 0.03905
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.08606 0.7692
## Lag[2*(p+q)+(p+q)-1][8] 1.45590 0.9323
## Lag[4*(p+q)+(p+q)-1][14] 3.48831 0.9239
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.8578 0.500 2.000 0.3544
## ARCH Lag[6] 1.5484 1.461 1.711 0.5982
## ARCH Lag[8] 2.4567 2.368 1.583 0.6492
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.4094
## Individual Statistics:
## omega 0.04653
## alpha1 0.04697
## beta1 0.02107
## beta2 0.04657
## gamma1 0.04634
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.1410 0.8884
## Negative Sign Bias 0.9332 0.3547
## Positive Sign Bias 0.2354 0.8148
## Joint Effect 2.0300 0.5662
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.10 0.15738
## 2 30 43.48 0.04105
## 3 40 38.65 0.48591
## 4 50 44.45 0.65783
##
##
## Elapsed time : 0.5517762egarch_21##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -1.743275 0.149016 -11.69854 0.00000
## alpha1 -0.094974 0.106868 -0.88870 0.37416
## alpha2 0.304436 0.195449 1.55763 0.11932
## beta1 0.587594 0.038619 15.21498 0.00000
## gamma1 -0.044683 0.338391 -0.13205 0.89495
## gamma2 -0.475254 0.309022 -1.53793 0.12406
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -1.743275 0.156695 -11.12528 0.000000
## alpha1 -0.094974 0.124402 -0.76345 0.445197
## alpha2 0.304436 0.144555 2.10603 0.035202
## beta1 0.587594 0.036735 15.99553 0.000000
## gamma1 -0.044683 0.316167 -0.14133 0.887611
## gamma2 -0.475254 0.346831 -1.37027 0.170601
##
## LogLikelihood : 39.6991
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.08707
## Bayes -0.88122
## Shibata -1.10369
## Hannan-Quinn -1.00625
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.681 0.1015
## Lag[2*(p+q)+(p+q)-1][2] 2.922 0.1473
## Lag[4*(p+q)+(p+q)-1][5] 4.357 0.2128
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04901 0.8248
## Lag[2*(p+q)+(p+q)-1][8] 3.44019 0.6049
## Lag[4*(p+q)+(p+q)-1][14] 8.76609 0.3033
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.01449 0.500 2.000 0.9042
## ARCH Lag[6] 2.54778 1.461 1.711 0.3816
## ARCH Lag[8] 3.62938 2.368 1.583 0.4330
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.9706
## Individual Statistics:
## omega 0.23381
## alpha1 0.10071
## alpha2 0.03783
## beta1 0.22477
## gamma1 0.40247
## gamma2 0.38356
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.6606 0.5115
## Negative Sign Bias 0.2729 0.7859
## Positive Sign Bias 0.4634 0.6449
## Joint Effect 0.4586 0.9279
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.13 0.7761
## 2 30 18.32 0.9374
## 3 40 28.32 0.8968
## 4 50 39.61 0.8285
##
##
## Elapsed time : 0.1773648egarch_13##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,3)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.49665 0.000053 -9449.9 0
## alpha1 -0.60972 0.000325 -1873.8 0
## beta1 0.26571 0.000059 4470.1 0
## beta2 -0.17054 0.000041 -4207.3 0
## beta3 0.86601 0.000175 4946.9 0
## gamma1 -1.53117 0.000342 -4477.4 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.49665 0.003222 -154.14 0
## alpha1 -0.60972 0.002817 -216.41 0
## beta1 0.26571 0.000828 321.02 0
## beta2 -0.17054 0.000703 -242.54 0
## beta3 0.86601 0.004238 204.36 0
## gamma1 -1.53117 0.008821 -173.59 0
##
## LogLikelihood : 44.82932
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.2526
## Bayes -1.0467
## Shibata -1.2692
## Hannan-Quinn -1.1717
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 8.083 0.004469
## Lag[2*(p+q)+(p+q)-1][2] 8.295 0.005403
## Lag[4*(p+q)+(p+q)-1][5] 9.181 0.014896
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 7.928 0.004868
## Lag[2*(p+q)+(p+q)-1][11] 11.131 0.056892
## Lag[4*(p+q)+(p+q)-1][19] 13.093 0.188452
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.09174 0.500 2.000 0.7620
## ARCH Lag[7] 1.02962 1.473 1.746 0.7490
## ARCH Lag[9] 1.44525 2.402 1.619 0.8616
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.7275
## Individual Statistics:
## omega 0.10584
## alpha1 0.10768
## beta1 0.03116
## beta2 0.10749
## beta3 0.01073
## gamma1 0.10628
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9695 3.364e-01
## Negative Sign Bias 2.1519 3.565e-02 **
## Positive Sign Bias 4.1381 1.167e-04 ***
## Joint Effect 21.7597 7.319e-05 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.16 0.2303
## 2 30 37.55 0.1328
## 3 40 46.13 0.2013
## 4 50 48.05 0.5116
##
##
## Elapsed time : 0.3322079egarch_31##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(3,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.920309 0.000220 -4179.20 0
## alpha1 -0.479239 0.000969 -494.35 0
## alpha2 0.633710 0.000272 2327.03 0
## alpha3 0.026277 0.000061 429.82 0
## beta1 0.800556 0.000114 7007.04 0
## gamma1 -0.664486 0.000485 -1369.00 0
## gamma2 -2.091609 0.000400 -5233.48 0
## gamma3 1.867172 0.000239 7808.64 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.920309 0.652466 -1.41051 0.158389
## alpha1 -0.479239 1.557952 -0.30761 0.758380
## alpha2 0.633710 0.063938 9.91136 0.000000
## alpha3 0.026277 0.074643 0.35203 0.724812
## beta1 0.800556 0.317097 2.52464 0.011582
## gamma1 -0.664486 0.262634 -2.53009 0.011403
## gamma2 -2.091609 1.570631 -1.33170 0.182959
## gamma3 1.867172 1.467327 1.27250 0.203196
##
## LogLikelihood : 51.50231
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.4033
## Bayes -1.1288
## Shibata -1.4318
## Hannan-Quinn -1.2955
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.396 0.1217
## Lag[2*(p+q)+(p+q)-1][2] 2.473 0.1951
## Lag[4*(p+q)+(p+q)-1][5] 3.435 0.3334
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06826 0.7939
## Lag[2*(p+q)+(p+q)-1][11] 6.92242 0.3215
## Lag[4*(p+q)+(p+q)-1][19] 12.79442 0.2073
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 1.462 0.500 2.000 0.2266
## ARCH Lag[7] 1.856 1.473 1.746 0.5365
## ARCH Lag[9] 3.888 2.402 1.619 0.4143
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.6022
## Individual Statistics:
## omega 0.02756
## alpha1 0.06821
## alpha2 0.06843
## alpha3 0.06838
## beta1 0.94555
## gamma1 0.06748
## gamma2 0.04101
## gamma3 0.02602
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.84033 0.07093 *
## Negative Sign Bias 0.06035 0.95209
## Positive Sign Bias 0.54484 0.58799
## Joint Effect 5.52819 0.13696
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 11.55 0.9040
## 2 30 22.19 0.8121
## 3 40 25.74 0.9493
## 4 50 43.21 0.7059
##
##
## Elapsed time : 1.138117egarch_11 Akaike -1.1560 Bayes -1.0188 Shibata -1.1637 Hannan-Quinn -1.1022
egarch_12 Akaike -1.2221 Bayes -1.0505 Shibata -1.2338 Hannan-Quinn -1.1547
egarch_21 Akaike -1.08707 Bayes -0.88122 Shibata -1.10369 Hannan-Quinn -1.00625
egarch_13 Akaike -1.2526 Bayes -1.0467 Shibata -1.2692 Hannan-Quinn -1.1717
egarch_31 Akaike -1.4033 Bayes -1.1288 Shibata -1.4318 Hannan-Quinn -1.2955
Na podstawie kryteriów informacyjnych wybieramy model eGARCH(3,1).
Wykres funkcji autokorelacji na kwadratach zwrotów:
plot(egarch_31, which = 10)
Wypustki są nieistotne, ponieważ nie przekraczają obszaru krytycznego. Zatem nie występuje autokorelacja reszt.
Kolejnym krokiem jest obliczenie Value-At_Risk w in sample i out of sample oraz porównanie kwantyli empirycznych ze standardowym z rozkładu normalnego.
portfel_probka_in$zwrotystd3 <- (portfel_probka_in$zwroty - mean(portfel_probka_in$zwroty, na.rm=T)) /
sd(portfel_probka_in$zwroty ,na.rm = T)
tail(portfel_probka_in)## Date zwroty zwrotystd VaR zwrotystd2 VaR2
## 116 2019-04-26 -0.02431955 -0.2332355 -0.3440883 -0.2332355 -0.3723686
## 117 2019-04-27 -0.01574112 -0.1692550 -0.3447191 -0.1692550 -0.3745176
## 118 2019-04-28 -0.06799414 -0.5589739 -0.3453481 -0.5589739 -0.3766487
## 119 2019-04-29 -0.16912923 -1.3132701 -0.3459754 -1.3132701 -0.3787623
## 120 2019-04-30 0.29343999 2.1367114 -0.3466009 2.1367114 -0.3808582
## 121 2019-05-01 -0.06367050 -0.5267269 -0.3472246 -0.5267269 -0.3829369
## zwrotystd3
## 116 -0.2332355
## 117 -0.1692550
## 118 -0.5589739
## 119 -1.3132701
## 120 2.1367114
## 121 -0.5267269basicStats(portfel_probka_in$zwrotystd3)## X..portfel_probka_in.zwrotystd3
## nobs 62.000000
## NAs 0.000000
## Minimum -2.831300
## Maximum 3.986365
## 1. Quartile -0.510065
## 3. Quartile 0.520829
## Mean 0.000000
## Median -0.059323
## Sum 0.000000
## SE Mean 0.127000
## LCL Mean -0.253952
## UCL Mean 0.253952
## Variance 1.000000
## Stdev 1.000000
## Skewness 0.632851
## Kurtosis 3.294645q01 <- quantile(portfel_probka_in$zwrotystd3, 0.01, na.rm = T)
q01## 1%
## -2.297398qnorm(0.01, 0, 1)## [1] -2.326348str(egarch_11)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:4, 1:4] 210390819 52471493 -24386658 -56804749 52471493 ...
## .. ..$ cvar : num [1:4, 1:4] -1.92e-10 1.90e-08 -2.66e-10 -6.05e-10 1.90e-08 ...
## .. ..$ var : num [1:62] 0.0177 0.0198 0.0219 0.0246 0.0243 ...
## .. ..$ sigma : num [1:62] 0.133 0.141 0.148 0.157 0.156 ...
## .. ..$ condH : num NaN
## .. ..$ z : num [1:62] 0.351 -0.358 -0.326 -0.536 0.968 ...
## .. ..$ LLH : num 39.8
## .. ..$ log.likelihoods: num [1:62] -1.036 -0.977 -0.939 -0.79 -0.472 ...
## .. ..$ residuals : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. ..$ coef : Named num [1:4] -0.2028 0.0111 0.99 -0.604
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. ..$ robust.cvar : num [1:4, 1:4] 1.00e-09 -1.32e-07 -1.98e-07 -4.39e-08 -1.32e-07 ...
## .. ..$ A : num [1:4, 1:4] 3393400 846314 -393333 -916206 846314 ...
## .. ..$ B : num [1:4, 1:4] 6389578 137865 -28177089 -2035575 137865 ...
## .. ..$ scores : num [1:62, 1:4] 0 0.436 0.938 1.172 0.151 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. ..$ se.coef : num [1:4] 1.39e-05 1.55e-04 2.64e-04 2.45e-04
## .. ..$ tval : Named num [1:4] -14626.7 71.5 3743.4 -2463
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. ..$ matcoef : num [1:4, 1:4] -2.03e-01 1.11e-02 9.90e-01 -6.04e-01 1.39e-05 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:4] 3.17e-05 4.16e-03 6.25e-03 1.39e-03
## .. ..$ robust.tval : Named num [1:4] -6401.99 2.66 158.37 -435.4
## .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. ..$ robust.matcoef : num [1:4, 1:4] -2.03e-01 1.11e-02 9.90e-01 -6.04e-01 3.17e-05 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:62] 0 0 0 0 0 0 0 0 0 0 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 0.798
## .. ..$ persistence : num 0.99
## .. ..$ timer : 'difftime' num 0.200273990631104
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:4] -0.2028 0.0111 0.99 -0.604
## .. .. .. .. ..- attr(*, "names")= chr [1:4] "omega" "alpha1" "beta1" "gamma1"
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:3] 26.8 -39.8 -39.8
## .. .. .. ..$ lagrange : num [1, 1] -19.3
## .. .. .. ..$ hessian : num [1:5, 1:5] 0.409 -9.948 -0.351 38.442 3.127 ...
## .. .. .. ..$ ineqx0 : Named num 1.01
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 346
## .. .. .. ..$ outer.iter : num 2
## .. .. .. ..$ elapsed : 'difftime' num 0.197808027267456
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:6] 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 0 0 0 0 0 0 1 1 1 1 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "eGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 0 0 0 0 0 0 1 2 3 4 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0egarch_11@fit$sigma## [1] 0.13317468 0.14081044 0.14789054 0.15678545 0.15573713 0.13691303
## [7] 0.13861630 0.14665302 0.15324952 0.14333067 0.15222139 0.13280632
## [13] 0.10981052 0.11778829 0.10383558 0.11826613 0.12657977 0.14043658
## [19] 0.14995027 0.16865121 0.18299938 0.16890772 0.18111240 0.17749405
## [25] 0.17861736 0.17233731 0.20118951 0.17972864 0.20998713 0.22709194
## [31] 0.25878785 0.27004206 0.27907832 0.18284635 0.17658659 0.20596295
## [37] 0.17623342 0.19014929 0.20032128 0.18882282 0.20804209 0.23431662
## [43] 0.16760901 0.15942699 0.16322303 0.16208049 0.13817030 0.13729482
## [49] 0.15033963 0.15182063 0.17589596 0.19677701 0.18584143 0.18770494
## [55] 0.21241861 0.17139729 0.13263305 0.14708081 0.16680423 0.17225059
## [61] 0.14903015 0.09741781portfel_probka_in$VaR3 <- q01 * egarch_11@fit$sigma
tail(portfel_probka_in)## Date zwroty zwrotystd VaR zwrotystd2 VaR2
## 116 2019-04-26 -0.02431955 -0.2332355 -0.3440883 -0.2332355 -0.3723686
## 117 2019-04-27 -0.01574112 -0.1692550 -0.3447191 -0.1692550 -0.3745176
## 118 2019-04-28 -0.06799414 -0.5589739 -0.3453481 -0.5589739 -0.3766487
## 119 2019-04-29 -0.16912923 -1.3132701 -0.3459754 -1.3132701 -0.3787623
## 120 2019-04-30 0.29343999 2.1367114 -0.3466009 2.1367114 -0.3808582
## 121 2019-05-01 -0.06367050 -0.5267269 -0.3472246 -0.5267269 -0.3829369
## zwrotystd3 VaR3
## 116 -0.2332355 -0.3047109
## 117 -0.1692550 -0.3379031
## 118 -0.5589739 -0.3832157
## 119 -1.3132701 -0.3957281
## 120 2.1367114 -0.3423815
## 121 -0.5267269 -0.2238074# wykres zwrotów i var
plot(portfel_probka_in$Date, portfel_probka_in$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-2,2))
abline(h = 0, lty = 2)
lines(portfel_probka_in$Date, portfel_probka_in$VaR3, type = 'l', col = "green")
a =sum(portfel_probka_in$zwroty < portfel_probka_in$VaR3) / length(portfel_probka_in$VaR3)
a## [1] 0W 0% przypadków straty przekroczyły zakładany poziom VaR.
VaR dla out of sample:
sigma.forecast3 <- ugarchforecast(egarch_11, n.ahead = 1)
str(sigma.forecast3)## Formal class 'uGARCHforecast' [package "rugarch"] with 2 slots
## ..@ forecast:List of 6
## .. ..$ n.ahead : num 1
## .. ..$ N : num 62
## .. ..$ n.start : num 0
## .. ..$ n.roll : num 0
## .. ..$ sigmaFor : num [1, 1] 0.0938
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## .. ..$ seriesFor: num [1, 1] 0
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr "T+1"
## .. .. .. ..$ : chr "1970-03-04 01:00:00"
## ..@ model :List of 11
## .. ..$ modelinc : Named num [1:22] 0 0 0 0 0 0 1 1 1 1 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "eGARCH"
## .. ..$ modeldata :List of 6
## .. .. ..$ T : int 62
## .. .. ..$ data : num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. .. ..$ index : POSIXct[1:62], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period : 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. .. ..$ sigma : num [1:62] 0.133 0.141 0.148 0.157 0.156 ...
## .. .. ..$ residuals: num [1:62] 0.0467 -0.0504 -0.0482 -0.084 0.1508 ...
## .. ..$ pars : num [1:19, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 1
## .. ..$ pos.matrix: num [1:21, 1:3] 0 0 0 0 0 0 1 2 3 4 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 9 10 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0str(sigma.forecast3@forecast)## List of 6
## $ n.ahead : num 1
## $ N : num 62
## $ n.start : num 0
## $ n.roll : num 0
## $ sigmaFor : num [1, 1] 0.0938
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"
## $ seriesFor: num [1, 1] 0
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr "T+1"
## .. ..$ : chr "1970-03-04 01:00:00"sigma.forecast3@forecast$sigmaFor## 1970-03-04 01:00:00
## T+1 0.09377283sigma.forecast33 <- sigma.forecast3@forecast$sigmaFor[1, 1]
# 1-dniowy VaR:
q01 * sigma.forecast33## 1%
## -0.2154335portfel$obs<-1:length(portfel$zwroty)
start <- portfel$obs[portfel$Date == as.Date("2019-03-02")]
finish <- portfel$obs[portfel$Date == as.Date("2019-06-01")]
portfel3 <-portfel[start:finish, ]
VaR3 <- rep(NA, times = finish - start + 1)
time1 <- Sys.time()
for (k in start:finish) {
tmp.data <- portfel[portfel$obs <= (k - 1), ]
tmp.data <- tmp.data[as.Date("2019-01-02") <= tmp.data$Date, ]
tmp.data$zwrotystd <- (tmp.data$zwroty - mean(tmp.data$zwroty, na.rm = T)) /
sd(tmp.data$zwroty, na.rm = T)
q01 <- quantile(tmp.data$zwrotystd, 0.01, na.rm = T)
espec11 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
tmp.egarch_13 <- ugarchfit(spec = espec_13, data = na.omit(portfel$zwroty))
sigma.forecast3 <- ugarchforecast(tmp.egarch_13, n.ahead = 1)
sigma.forecast33 <- sigma.forecast3@forecast$sigmaFor[1, 1]
VaR3[k - start + 1] <- q01 * sigma.forecast33
}
time2 <- Sys.time()
time2 - time1## Time difference of 20.01943 secsportfel3$VaR3 <- VaR3
head(portfel3)## Date zwroty obs VaR3
## 61 2019-03-02 -0.05042352 61 -0.3896493
## 62 2019-03-03 -0.04822774 62 -0.3906934
## 63 2019-03-04 -0.08402545 63 -0.3917885
## 64 2019-03-05 0.15076191 64 -0.3915307
## 65 2019-03-06 -0.06548814 65 -0.3921967
## 66 2019-03-07 0.04815102 66 -0.3925820tail(portfel3)## Date zwroty obs VaR3
## 147 2019-05-27 0.19794321 147 -0.3396306
## 148 2019-05-28 -0.00585839 148 -0.3400340
## 149 2019-05-29 0.03274560 149 -0.3410705
## 150 2019-05-30 -0.26059968 150 -0.3423246
## 151 2019-05-31 0.18577773 151 -0.3379694
## 152 2019-06-01 0.09272815 152 -0.3385231plot(portfel3$Date, portfel3$zwroty, col = "red", lwd = 1, type = 'l',
ylim = c(-1, 1))
abline(h = 0, lty = 2)
lines(portfel3$Date, portfel3$VaR3, type = 'l', col = "green")
b = sum(portfel3$zwroty < portfel3$VaR3) / length(portfel3$VaR3)
b## [1] 0W 0% przypadków straty przekroczyły zakładany poziom VaR.
Porównanie oszacowań wybranych modeli:
garch_11 ##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.017410 0.448338 0.65391
## omega 0.000105 0.001974 0.053073 0.95767
## alpha1 0.000000 0.048064 0.000000 1.00000
## beta1 0.999000 0.127840 7.814467 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.007806 0.011901 0.655892 0.511894
## omega 0.000105 0.003070 0.034119 0.972782
## alpha1 0.000000 0.114887 0.000000 1.000000
## beta1 0.999000 0.242399 4.121296 0.000038
##
## LogLikelihood : 37.53098
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.08164
## Bayes -0.94441
## Shibata -1.08932
## Hannan-Quinn -1.02776
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.332 0.2484
## Lag[2*(p+q)+(p+q)-1][2] 1.379 0.3901
## Lag[4*(p+q)+(p+q)-1][5] 2.567 0.4917
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01737 0.8951
## Lag[2*(p+q)+(p+q)-1][5] 0.75792 0.9116
## Lag[4*(p+q)+(p+q)-1][9] 2.60867 0.8214
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.07056 0.500 2.000 0.7905
## ARCH Lag[5] 0.45740 1.440 1.667 0.8961
## ARCH Lag[7] 1.19966 2.315 1.543 0.8791
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.6032
## Individual Statistics:
## mu 0.16164
## omega 0.05206
## alpha1 0.05538
## beta1 0.04945
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.2212 0.8257
## Negative Sign Bias 0.1040 0.9175
## Positive Sign Bias 0.1687 0.8666
## Joint Effect 0.0537 0.9967
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 14.77 0.7368
## 2 30 26.06 0.6220
## 3 40 27.03 0.9262
## 4 50 46.06 0.5929
##
##
## Elapsed time : 0.10078k.tgarch_11 ##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002279 0.004699 0.485102 0.6276
## alpha1 0.000000 0.000002 0.000131 0.9999
## beta1 0.991708 0.038038 26.071406 0.0000
## eta11 -0.985951 1.677084 -0.587896 0.5566
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.002279 0.004043 0.56384 0.57286
## alpha1 0.000000 0.000000 0.30145 0.76307
## beta1 0.991708 0.029365 33.77200 0.00000
## eta11 -0.985951 4.316444 -0.22842 0.81932
##
## LogLikelihood : 39.61836
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.1490
## Bayes -1.0117
## Shibata -1.1567
## Hannan-Quinn -1.0951
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.806 0.1790
## Lag[2*(p+q)+(p+q)-1][2] 1.865 0.2862
## Lag[4*(p+q)+(p+q)-1][5] 2.986 0.4097
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06796 0.7943
## Lag[2*(p+q)+(p+q)-1][5] 0.86983 0.8884
## Lag[4*(p+q)+(p+q)-1][9] 2.68274 0.8100
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.08566 0.500 2.000 0.7698
## ARCH Lag[5] 0.70240 1.440 1.667 0.8227
## ARCH Lag[7] 1.50849 2.315 1.543 0.8197
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.2742
## Individual Statistics:
## omega 0.05682
## alpha1 0.05023
## beta1 0.05659
## eta11 0.06104
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.4090 0.6841
## Negative Sign Bias 0.1023 0.9189
## Positive Sign Bias 0.3188 0.7511
## Joint Effect 0.1904 0.9791
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.29 0.4384
## 2 30 23.16 0.7691
## 3 40 33.48 0.7191
## 4 50 34.77 0.9378
##
##
## Elapsed time : 0.1547849egarch_31##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(3,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.920309 0.000220 -4179.20 0
## alpha1 -0.479239 0.000969 -494.35 0
## alpha2 0.633710 0.000272 2327.03 0
## alpha3 0.026277 0.000061 429.82 0
## beta1 0.800556 0.000114 7007.04 0
## gamma1 -0.664486 0.000485 -1369.00 0
## gamma2 -2.091609 0.000400 -5233.48 0
## gamma3 1.867172 0.000239 7808.64 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.920309 0.652466 -1.41051 0.158389
## alpha1 -0.479239 1.557952 -0.30761 0.758380
## alpha2 0.633710 0.063938 9.91136 0.000000
## alpha3 0.026277 0.074643 0.35203 0.724812
## beta1 0.800556 0.317097 2.52464 0.011582
## gamma1 -0.664486 0.262634 -2.53009 0.011403
## gamma2 -2.091609 1.570631 -1.33170 0.182959
## gamma3 1.867172 1.467327 1.27250 0.203196
##
## LogLikelihood : 51.50231
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.4033
## Bayes -1.1288
## Shibata -1.4318
## Hannan-Quinn -1.2955
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.396 0.1217
## Lag[2*(p+q)+(p+q)-1][2] 2.473 0.1951
## Lag[4*(p+q)+(p+q)-1][5] 3.435 0.3334
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06826 0.7939
## Lag[2*(p+q)+(p+q)-1][11] 6.92242 0.3215
## Lag[4*(p+q)+(p+q)-1][19] 12.79442 0.2073
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 1.462 0.500 2.000 0.2266
## ARCH Lag[7] 1.856 1.473 1.746 0.5365
## ARCH Lag[9] 3.888 2.402 1.619 0.4143
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.6022
## Individual Statistics:
## omega 0.02756
## alpha1 0.06821
## alpha2 0.06843
## alpha3 0.06838
## beta1 0.94555
## gamma1 0.06748
## gamma2 0.04101
## gamma3 0.02602
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.84033 0.07093 *
## Negative Sign Bias 0.06035 0.95209
## Positive Sign Bias 0.54484 0.58799
## Joint Effect 5.52819 0.13696
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 11.55 0.9040
## 2 30 22.19 0.8121
## 3 40 25.74 0.9493
## 4 50 43.21 0.7059
##
##
## Elapsed time : 1.138117Spośród wybranych modeli najmniejszą wartość kryteriów informacyjnych ma eGARCH(3,1).
#Prognozowanie wariancji bezwarunkowej dla eGARCH(3,1)
egarch_31 <- garchFit(formula = ~ garch(3, 1),
data = na.omit(portfel$zwroty),
include.mean = F,
cond.dist = "norm",
trace = F)
egarch_31@fit$par## omega alpha1 alpha2 alpha3 beta1
## 0.006961535 0.031678286 0.000000010 0.126104199 0.574852985var_uncond <- egarch_31@fit$par[1] / (1 - egarch_31@fit$par[2]
- egarch_31@fit$par[3])
names(var_uncond) <- "unconditional variance egarch"
var_uncond## unconditional variance egarch
## 0.007189279fore31 <- predict(egarch_31, n.ahead = 20)
head(fore31)## meanForecast meanError standardDeviation
## 1 0 0.1404761 0.1404761
## 2 0 0.1376984 0.1376984
## 3 0 0.1381412 0.1381412
## 4 0 0.1449981 0.1449981
## 5 0 0.1486760 0.1486760
## 6 0 0.1509142 0.1509142plot(fore31[, 3] ^ 2, type = "l")
abline(h = var_uncond, col = "red", lty = 2)
title(main = "Warunkowa i bezwarunkowa wariancja zwrotów eGARCH(3,1)")
# Wykres wariancji out of sample GARCH(1,1)
spec_11 <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0),
include.mean = T),
distribution.model = "norm")
garch_11 <- ugarchfit(spec = spec_11,
data = na.omit(portfel$zwroty))
garch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.005311 0.009586 -0.55405 0.579541
## omega 0.004206 0.002560 1.64307 0.100369
## alpha1 0.097716 0.055060 1.77473 0.075943
## beta1 0.741153 0.131971 5.61603 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.005311 0.008305 -0.63952 0.522487
## omega 0.004206 0.001982 2.12263 0.033785
## alpha1 0.097716 0.055143 1.77205 0.076387
## beta1 0.741153 0.090549 8.18515 0.000000
##
## LogLikelihood : 104.2212
##
## Information Criteria
## ------------------------------------
##
## Akaike -0.81481
## Bayes -0.75781
## Shibata -0.81532
## Hannan-Quinn -0.79186
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.972 0.1603
## Lag[2*(p+q)+(p+q)-1][2] 2.622 0.1777
## Lag[4*(p+q)+(p+q)-1][5] 3.133 0.3833
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.03868 0.8441
## Lag[2*(p+q)+(p+q)-1][5] 2.12420 0.5894
## Lag[4*(p+q)+(p+q)-1][9] 3.16173 0.7320
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.000784 0.500 2.000 0.9777
## ARCH Lag[5] 2.308383 1.440 1.667 0.4070
## ARCH Lag[7] 2.610566 2.315 1.543 0.5899
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.5333
## Individual Statistics:
## mu 0.2432
## omega 0.1595
## alpha1 0.1900
## beta1 0.1834
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.5045 0.1338
## Negative Sign Bias 1.1499 0.2513
## Positive Sign Bias 0.7339 0.4637
## Joint Effect 2.4580 0.4829
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 32.86 0.02494
## 2 30 41.32 0.06460
## 3 40 51.56 0.08584
## 4 50 49.12 0.46824
##
##
## Elapsed time : 0.0890789plot(ugarchforecast(garch_11, n.ahead=20), which = 3)
# Wykres wariancji out of sample eGARCH(3,1)
espec_31 = ugarchspec(variance.model = list(model ="eGARCH",
garchOrder = c(3, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = F),
distribution.model = "norm")
egarch_31 <- ugarchfit(spec = espec_31, data = na.omit(portfel$zwroty))
egarch_31##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(3,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.855075 0.486417 -1.75790 0.078764
## alpha1 -0.028408 0.089092 -0.31886 0.749830
## alpha2 0.043584 0.115669 0.37680 0.706323
## alpha3 0.039053 0.117465 0.33247 0.739536
## beta1 0.762994 0.132865 5.74264 0.000000
## gamma1 0.168874 0.153801 1.09800 0.272205
## gamma2 -0.105016 0.162740 -0.64530 0.518734
## gamma3 0.258664 0.189544 1.36467 0.172358
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.855075 0.340554 -2.51084 0.012045
## alpha1 -0.028408 0.092998 -0.30547 0.760008
## alpha2 0.043584 0.117257 0.37169 0.710120
## alpha3 0.039053 0.125862 0.31029 0.756343
## beta1 0.762994 0.094801 8.04834 0.000000
## gamma1 0.168874 0.151188 1.11698 0.264003
## gamma2 -0.105016 0.147124 -0.71379 0.475356
## gamma3 0.258664 0.197735 1.30813 0.190829
##
## LogLikelihood : 106.1957
##
## Information Criteria
## ------------------------------------
##
## Akaike -0.79834
## Bayes -0.68434
## Shibata -0.80037
## Hannan-Quinn -0.75244
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.626 0.1051
## Lag[2*(p+q)+(p+q)-1][2] 3.251 0.1200
## Lag[4*(p+q)+(p+q)-1][5] 3.757 0.2860
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.05428 0.8158
## Lag[2*(p+q)+(p+q)-1][11] 1.60387 0.9809
## Lag[4*(p+q)+(p+q)-1][19] 2.79822 0.9966
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.1230 0.500 2.000 0.7258
## ARCH Lag[7] 0.4655 1.473 1.746 0.9064
## ARCH Lag[9] 0.5633 2.402 1.619 0.9792
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.153
## Individual Statistics:
## omega 0.11240
## alpha1 0.17869
## alpha2 0.07197
## alpha3 0.08578
## beta1 0.10303
## gamma1 0.08577
## gamma2 0.11828
## gamma3 0.11169
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0622 0.04026 **
## Negative Sign Bias 1.2585 0.20943
## Positive Sign Bias 0.9056 0.36604
## Joint Effect 4.3377 0.22723
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 28.31 0.07766
## 2 30 41.32 0.06460
## 3 40 42.46 0.32445
## 4 50 61.32 0.11140
##
##
## Elapsed time : 0.1796088str(egarch_31)## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
## ..@ fit :List of 27
## .. ..$ hessian : num [1:8, 1:8] 982.53 -9.22 -28.89 -25.33 -3650.39 ...
## .. ..$ cvar : num [1:8, 1:8] 0.2366 0.00608 -0.00163 0.00666 0.06447 ...
## .. ..$ var : num [1:246] 0.0264 0.0264 0.0264 0.0221 0.0217 ...
## .. ..$ sigma : num [1:246] 0.162 0.162 0.162 0.149 0.147 ...
## .. ..$ condH : num 3.59
## .. ..$ z : num [1:246] 0.409 -0.735 0.428 0.413 1.674 ...
## .. ..$ LLH : num 106
## .. ..$ log.likelihoods: num [1:246] -0.815 -0.629 -0.807 -0.902 0.406 ...
## .. ..$ residuals : num [1:246] 0.0664 -0.1193 0.0695 0.0614 0.2468 ...
## .. ..$ coef : Named num [1:8] -0.8551 -0.0284 0.0436 0.0391 0.763 ...
## .. .. ..- attr(*, "names")= chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. ..$ robust.cvar : num [1:8, 1:8] 0.115977 0.008269 -0.000561 0.005163 0.031999 ...
## .. ..$ A : num [1:8, 1:8] 3.994 -0.0375 -0.1174 -0.103 -14.839 ...
## .. ..$ B : num [1:8, 1:8] 7.582 -0.202 -0.473 -0.24 -28.331 ...
## .. ..$ scores : num [1:246, 1:8] 0 0 0 0.415 -1.562 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : NULL
## .. .. .. ..$ : chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. ..$ se.coef : num [1:8] 0.4864 0.0891 0.1157 0.1175 0.1329 ...
## .. ..$ tval : Named num [1:8] -1.758 -0.319 0.377 0.332 5.743 ...
## .. .. ..- attr(*, "names")= chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. ..$ matcoef : num [1:8, 1:4] -0.8551 -0.0284 0.0436 0.0391 0.763 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ robust.se.coef : num [1:8] 0.3406 0.093 0.1173 0.1259 0.0948 ...
## .. ..$ robust.tval : Named num [1:8] -2.511 -0.305 0.372 0.31 8.048 ...
## .. .. ..- attr(*, "names")= chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. ..$ robust.matcoef : num [1:8, 1:4] -0.8551 -0.0284 0.0436 0.0391 0.763 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
## .. ..$ fitted.values : num [1:246] 0 0 0 0 0 0 0 0 0 0 ...
## .. ..$ convergence : num 0
## .. ..$ kappa : num 0.798
## .. ..$ persistence : num 0.763
## .. ..$ timer : 'difftime' num 0.179608821868896
## .. .. ..- attr(*, "units")= chr "secs"
## .. ..$ ipars : num [1:23, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:23] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ solver :List of 2
## .. .. ..$ sol :List of 10
## .. .. .. ..$ pars : Named num [1:8] -0.8551 -0.0284 0.0436 0.0391 0.763 ...
## .. .. .. .. ..- attr(*, "names")= chr [1:8] "omega" "alpha1" "alpha2" "alpha3" ...
## .. .. .. ..$ convergence: num 0
## .. .. .. ..$ values : num [1:4] 168 -106 -106 -106
## .. .. .. ..$ lagrange : num [1, 1] 4.2e-07
## .. .. .. ..$ hessian : num [1:9, 1:9] 0.3988 -0.0967 -0.1442 -0.5164 -0.1248 ...
## .. .. .. ..$ ineqx0 : Named num 1.31
## .. .. .. .. ..- attr(*, "names")= chr ""
## .. .. .. ..$ nfuneval : num 296
## .. .. .. ..$ outer.iter : num 3
## .. .. .. ..$ elapsed : 'difftime' num 0.177961111068726
## .. .. .. .. ..- attr(*, "units")= chr "secs"
## .. .. .. ..$ vscale : num [1:10] 1 1 1 1 1 1 1 1 1 1
## .. .. ..$ hess: NULL
## ..@ model:List of 11
## .. ..$ modelinc : Named num [1:22] 0 0 0 0 0 0 1 3 1 3 ...
## .. .. ..- attr(*, "names")= chr [1:22] "mu" "ar" "ma" "arfima" ...
## .. ..$ modeldesc :List of 3
## .. .. ..$ distribution: chr "norm"
## .. .. ..$ distno : int 1
## .. .. ..$ vmodel : chr "eGARCH"
## .. ..$ modeldata :List of 4
## .. .. ..$ T : int 246
## .. .. ..$ data : num [1:246] 0.0664 -0.1193 0.0695 0.0614 0.2468 ...
## .. .. ..$ index : POSIXct[1:246], format: "1970-01-02 01:00:00" ...
## .. .. ..$ period: 'difftime' num 1
## .. .. .. ..- attr(*, "units")= chr "days"
## .. ..$ pars : num [1:23, 1:6] 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:23] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:6] "Level" "Fixed" "Include" "Estimate" ...
## .. ..$ start.pars: NULL
## .. ..$ fixed.pars: NULL
## .. ..$ maxOrder : num 3
## .. ..$ pos.matrix: num [1:21, 1:3] 0 0 0 0 0 0 1 2 5 6 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:21] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:3] "start" "stop" "include"
## .. ..$ fmodel : NULL
## .. ..$ pidx : num [1:19, 1:2] 1 2 3 4 5 6 7 8 11 12 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:19] "mu" "ar" "ma" "arfima" ...
## .. .. .. ..$ : chr [1:2] "begin" "end"
## .. ..$ n.start : num 0plot(ugarchforecast(egarch_31, n.ahead=20), which = 3)
# Wykres wariancji out of sample tGARCH(1,1)
tspec_11 <- ugarchspec(variance.model = list(model = "fGARCH",
garchOrder = c(1, 1),
submodel = "TGARCH"),
mean.model = list(armaOrder = c(0, 0),
include.mean = F),
distribution.model = "norm")
k.tgarch_11 <- ugarchfit(spec = tspec_11, data = na.omit(portfel$zwroty))
k.tgarch_11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000857 0.001332 0.64349 0.519906
## alpha1 0.038490 0.009125 4.21834 0.000025
## beta1 0.967384 0.022337 43.30859 0.000000
## eta11 -1.000000 0.464118 -2.15462 0.031191
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000857 0.003340 0.25661 0.797478
## alpha1 0.038490 0.020511 1.87655 0.060579
## beta1 0.967384 0.028911 33.46113 0.000000
## eta11 -1.000000 0.664556 -1.50476 0.132385
##
## LogLikelihood : 103.3991
##
## Information Criteria
## ------------------------------------
##
## Akaike -0.80812
## Bayes -0.75113
## Shibata -0.80864
## Hannan-Quinn -0.78517
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.601 0.05776
## Lag[2*(p+q)+(p+q)-1][2] 4.123 0.06988
## Lag[4*(p+q)+(p+q)-1][5] 4.858 0.16482
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2414 0.6232
## Lag[2*(p+q)+(p+q)-1][5] 2.5203 0.5016
## Lag[4*(p+q)+(p+q)-1][9] 4.3413 0.5320
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1146 0.500 2.000 0.7350
## ARCH Lag[5] 4.4293 1.440 1.667 0.1390
## ARCH Lag[7] 5.1382 2.315 1.543 0.2109
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.3099
## Individual Statistics:
## omega 0.08987
## alpha1 0.09071
## beta1 0.09905
## eta11 0.06369
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.8285 0.06871 *
## Negative Sign Bias 1.8698 0.06272 *
## Positive Sign Bias 0.5644 0.57304
## Joint Effect 4.5618 0.20684
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 28.15 0.08065
## 2 30 36.44 0.16112
## 3 40 49.28 0.12517
## 4 50 63.76 0.07656
##
##
## Elapsed time : 0.1639779plot(ugarchforecast(k.tgarch_11, n.ahead=20), which = 3)
Najlepszym z powyższych modeli jest model eGARCH(3,1). Wszystkie oszacowane parametry modelu są istotne.